id sid tid token lemma pos ejde-67 1 1 electronic electronic ADJ ejde-67 1 2 journal journal NOUN ejde-67 1 3 of of ADP ejde-67 1 4 differential differential ADJ ejde-67 1 5 equations equation NOUN ejde-67 1 6 , , PUNCT ejde-67 1 7 vol vol NOUN ejde-67 1 8 . . PUNCT ejde-67 1 9 2023 2023 NUM ejde-67 1 10 ( ( PUNCT ejde-67 1 11 2023 2023 NUM ejde-67 1 12 ) ) PUNCT ejde-67 1 13 , , PUNCT ejde-67 1 14 no no INTJ ejde-67 1 15 . . NOUN ejde-67 1 16 39 39 NUM ejde-67 1 17 , , PUNCT ejde-67 1 18 pp pp ADJ ejde-67 1 19 . . PUNCT ejde-67 2 1 1–17 1–17 NOUN ejde-67 2 2 . . PUNCT ejde-67 3 1 issn issn PROPN ejde-67 3 2 : : PUNCT ejde-67 3 3 1072 1072 NUM ejde-67 3 4 - - SYM ejde-67 3 5 6691 6691 NUM ejde-67 3 6 . . PUNCT ejde-67 4 1 url url PROPN ejde-67 4 2 : : PUNCT ejde-67 4 3 https://ejde.math.txstate.edu https://ejde.math.txstate.edu PROPN ejde-67 4 4 , , PUNCT ejde-67 4 5 https://ejde.math.unt.edu https://ejde.math.unt.edu PROPN ejde-67 4 6 doi doi PROPN ejde-67 4 7 : : PUNCT ejde-67 4 8 https://doi.org/10.58997/ejde.2023.39 https://doi.org/10.58997/ejde.2023.39 VERB ejde-67 4 9 reduction reduction NOUN ejde-67 4 10 principle principle NOUN ejde-67 4 11 for for ADP ejde-67 4 12 partial partial ADJ ejde-67 4 13 functional functional ADJ ejde-67 4 14 differential differential NOUN ejde-67 4 15 equation equation NOUN ejde-67 4 16 without without ADP ejde-67 4 17 compactness compactness NOUN ejde-67 4 18 meryem meryem PROPN ejde-67 4 19 el el PROPN ejde-67 4 20 attaouy attaouy PROPN ejde-67 4 21 , , PUNCT ejde-67 4 22 khalil khalil PROPN ejde-67 4 23 ezzinbi ezzinbi PROPN ejde-67 4 24 , , PUNCT ejde-67 4 25 gaston gaston PROPN ejde-67 4 26 mandata mandata PROPN ejde-67 4 27 n’guérékata n’guérékata PROPN ejde-67 4 28 abstract abstract PROPN ejde-67 4 29 . . PUNCT ejde-67 5 1 this this DET ejde-67 5 2 article article NOUN ejde-67 5 3 establishes establish VERB ejde-67 5 4 a a DET ejde-67 5 5 reduction reduction NOUN ejde-67 5 6 principle principle NOUN ejde-67 5 7 for for ADP ejde-67 5 8 partial partial ADJ ejde-67 5 9 functional functional ADJ ejde-67 5 10 differential differential NOUN ejde-67 5 11 equation equation NOUN ejde-67 5 12 without without ADP ejde-67 5 13 compactness compactness NOUN ejde-67 5 14 of of ADP ejde-67 5 15 the the DET ejde-67 5 16 semigroup semigroup NOUN ejde-67 5 17 generated generate VERB ejde-67 5 18 by by ADP ejde-67 5 19 the the DET ejde-67 5 20 linear linear ADJ ejde-67 5 21 part part NOUN ejde-67 5 22 . . PUNCT ejde-67 6 1 under under ADP ejde-67 6 2 conditions condition NOUN ejde-67 6 3 more more ADV ejde-67 6 4 general general ADJ ejde-67 6 5 than than ADP ejde-67 6 6 the the DET ejde-67 6 7 compactness compactness NOUN ejde-67 6 8 of of ADP ejde-67 6 9 the the DET ejde-67 6 10 c0semigroup c0semigroup NOUN ejde-67 6 11 generated generate VERB ejde-67 6 12 by by ADP ejde-67 6 13 the the DET ejde-67 6 14 linear linear ADJ ejde-67 6 15 part part NOUN ejde-67 6 16 , , PUNCT ejde-67 6 17 we we PRON ejde-67 6 18 establish establish VERB ejde-67 6 19 the the DET ejde-67 6 20 quasi quasi NOUN ejde-67 6 21 - - NOUN ejde-67 6 22 compactness compactness NOUN ejde-67 6 23 of of ADP ejde-67 6 24 the the DET ejde-67 6 25 c0 c0 PROPN ejde-67 6 26 - - PUNCT ejde-67 6 27 semigroup semigroup PROPN ejde-67 6 28 associated associate VERB ejde-67 6 29 to to ADP ejde-67 6 30 the the DET ejde-67 6 31 linear linear ADJ ejde-67 6 32 part part NOUN ejde-67 6 33 of of ADP ejde-67 6 34 the the DET ejde-67 6 35 partial partial ADJ ejde-67 6 36 functional functional ADJ ejde-67 6 37 differential differential NOUN ejde-67 6 38 equation equation NOUN ejde-67 6 39 . . PUNCT ejde-67 7 1 this this DET ejde-67 7 2 result result NOUN ejde-67 7 3 allows allow VERB ejde-67 7 4 as as SCONJ ejde-67 7 5 to to PART ejde-67 7 6 construct construct VERB ejde-67 7 7 a a DET ejde-67 7 8 reduced reduce VERB ejde-67 7 9 system system NOUN ejde-67 7 10 that that PRON ejde-67 7 11 is be AUX ejde-67 7 12 posed pose VERB ejde-67 7 13 by by ADP ejde-67 7 14 an an DET ejde-67 7 15 ordinary ordinary ADJ ejde-67 7 16 differential differential ADJ ejde-67 7 17 equation equation NOUN ejde-67 7 18 posed pose VERB ejde-67 7 19 in in ADP ejde-67 7 20 a a DET ejde-67 7 21 finite finite ADJ ejde-67 7 22 dimensional dimensional ADJ ejde-67 7 23 space space NOUN ejde-67 7 24 . . PUNCT ejde-67 8 1 through through ADP ejde-67 8 2 this this DET ejde-67 8 3 result result NOUN ejde-67 8 4 we we PRON ejde-67 8 5 study study VERB ejde-67 8 6 the the DET ejde-67 8 7 existence existence NOUN ejde-67 8 8 of of ADP ejde-67 8 9 almost almost ADV ejde-67 8 10 automorphic automorphic ADJ ejde-67 8 11 and and CCONJ ejde-67 8 12 almost almost ADV ejde-67 8 13 periodic periodic ADJ ejde-67 8 14 solutions solution NOUN ejde-67 8 15 for for ADP ejde-67 8 16 partial partial ADJ ejde-67 8 17 functional functional ADJ ejde-67 8 18 differential differential NOUN ejde-67 8 19 equations equation NOUN ejde-67 8 20 . . PUNCT ejde-67 9 1 for for ADP ejde-67 9 2 illustration illustration NOUN ejde-67 9 3 , , PUNCT ejde-67 9 4 we we PRON ejde-67 9 5 study study VERB ejde-67 9 6 a a DET ejde-67 9 7 transport transport NOUN ejde-67 9 8 model model NOUN ejde-67 9 9 . . PUNCT ejde-67 10 1 1 1 X ejde-67 10 2 . . X ejde-67 10 3 introduction introduction NOUN ejde-67 10 4 the the DET ejde-67 10 5 theory theory NOUN ejde-67 10 6 of of ADP ejde-67 10 7 functional functional ADJ ejde-67 10 8 differential differential ADJ ejde-67 10 9 equations equation NOUN ejde-67 10 10 with with ADP ejde-67 10 11 delay delay NOUN ejde-67 10 12 has have AUX ejde-67 10 13 emerged emerge VERB ejde-67 10 14 as as ADP ejde-67 10 15 an an DET ejde-67 10 16 important important ADJ ejde-67 10 17 branch branch NOUN ejde-67 10 18 of of ADP ejde-67 10 19 nonlinear nonlinear ADJ ejde-67 10 20 analysis analysis NOUN ejde-67 10 21 because because SCONJ ejde-67 10 22 it it PRON ejde-67 10 23 has have VERB ejde-67 10 24 wide wide ADJ ejde-67 10 25 range range NOUN ejde-67 10 26 of of ADP ejde-67 10 27 application application NOUN ejde-67 10 28 in in ADP ejde-67 10 29 various various ADJ ejde-67 10 30 fields field NOUN ejde-67 10 31 of of ADP ejde-67 10 32 pure pure ADJ ejde-67 10 33 and and CCONJ ejde-67 10 34 applied applied ADJ ejde-67 10 35 mathematics mathematic NOUN ejde-67 10 36 as as ADV ejde-67 10 37 well well ADV ejde-67 10 38 as as ADP ejde-67 10 39 in in ADP ejde-67 10 40 other other ADJ ejde-67 10 41 fields field NOUN ejde-67 10 42 like like ADP ejde-67 10 43 physics physics NOUN ejde-67 10 44 , , PUNCT ejde-67 10 45 chemistry chemistry NOUN ejde-67 10 46 , , PUNCT ejde-67 10 47 population population NOUN ejde-67 10 48 dynamics dynamic NOUN ejde-67 10 49 , , PUNCT ejde-67 10 50 biology biology NOUN ejde-67 10 51 , , PUNCT ejde-67 10 52 engineering engineering NOUN ejde-67 10 53 , , PUNCT ejde-67 10 54 economics economic NOUN ejde-67 10 55 , , PUNCT ejde-67 10 56 and and CCONJ ejde-67 10 57 so so ADV ejde-67 10 58 on on ADV ejde-67 10 59 . . PUNCT ejde-67 11 1 one one NUM ejde-67 11 2 of of ADP ejde-67 11 3 the the DET ejde-67 11 4 theories theory NOUN ejde-67 11 5 related relate VERB ejde-67 11 6 to to ADP ejde-67 11 7 functional functional ADJ ejde-67 11 8 differential differential ADJ ejde-67 11 9 equations equation NOUN ejde-67 11 10 with with ADP ejde-67 11 11 delay delay NOUN ejde-67 11 12 is be AUX ejde-67 11 13 the the DET ejde-67 11 14 one one NUM ejde-67 11 15 of of ADP ejde-67 11 16 almost almost ADV ejde-67 11 17 automorphy automorphy NOUN ejde-67 11 18 . . PUNCT ejde-67 12 1 this this DET ejde-67 12 2 last last ADJ ejde-67 12 3 notion notion NOUN ejde-67 12 4 has have AUX ejde-67 12 5 been be AUX ejde-67 12 6 introduced introduce VERB ejde-67 12 7 by by ADP ejde-67 12 8 bochner bochner NOUN ejde-67 12 9 in in ADP ejde-67 12 10 1950 1950 NUM ejde-67 12 11 , , PUNCT ejde-67 12 12 as as ADP ejde-67 12 13 a a DET ejde-67 12 14 generalization generalization NOUN ejde-67 12 15 of of ADP ejde-67 12 16 almost almost ADV ejde-67 12 17 periodicity periodicity NOUN ejde-67 12 18 [ [ X ejde-67 12 19 3 3 NUM ejde-67 12 20 ] ] PUNCT ejde-67 12 21 . . PUNCT ejde-67 13 1 the the DET ejde-67 13 2 problem problem NOUN ejde-67 13 3 of of ADP ejde-67 13 4 the the DET ejde-67 13 5 existence existence NOUN ejde-67 13 6 of of ADP ejde-67 13 7 periodic periodic ADJ ejde-67 13 8 and and CCONJ ejde-67 13 9 almost almost ADV ejde-67 13 10 periodic periodic ADJ ejde-67 13 11 solutions solution NOUN ejde-67 13 12 of of ADP ejde-67 13 13 functional functional ADJ ejde-67 13 14 differential differential ADJ ejde-67 13 15 equations equation NOUN ejde-67 13 16 with with ADP ejde-67 13 17 delay delay NOUN ejde-67 13 18 have have AUX ejde-67 13 19 received receive VERB ejde-67 13 20 the the DET ejde-67 13 21 attention attention NOUN ejde-67 13 22 of of ADP ejde-67 13 23 many many ADJ ejde-67 13 24 authors author NOUN ejde-67 13 25 . . PUNCT ejde-67 14 1 we we PRON ejde-67 14 2 refer refer VERB ejde-67 14 3 the the DET ejde-67 14 4 reader reader NOUN ejde-67 14 5 to to ADP ejde-67 14 6 the the DET ejde-67 14 7 book book NOUN ejde-67 14 8 [ [ X ejde-67 14 9 13 13 NUM ejde-67 14 10 ] ] PUNCT ejde-67 14 11 and and CCONJ ejde-67 14 12 to to ADP ejde-67 14 13 the the DET ejde-67 14 14 papers paper NOUN ejde-67 14 15 [ [ X ejde-67 14 16 5 5 NUM ejde-67 14 17 , , PUNCT ejde-67 14 18 20 20 NUM ejde-67 14 19 ] ] PUNCT ejde-67 14 20 . . PUNCT ejde-67 15 1 more more ADV ejde-67 15 2 recently recently ADV ejde-67 15 3 , , PUNCT ejde-67 15 4 the the DET ejde-67 15 5 existence existence NOUN ejde-67 15 6 of of ADP ejde-67 15 7 almost almost ADV ejde-67 15 8 automorphic automorphic ADJ ejde-67 15 9 solutions solution NOUN ejde-67 15 10 to to ADP ejde-67 15 11 ordinary ordinary ADJ ejde-67 15 12 as as ADV ejde-67 15 13 well well ADV ejde-67 15 14 as as ADP ejde-67 15 15 abstract abstract ADJ ejde-67 15 16 differential differential ADJ ejde-67 15 17 equations equation NOUN ejde-67 15 18 has have AUX ejde-67 15 19 been be AUX ejde-67 15 20 intensively intensively ADV ejde-67 15 21 studied study VERB ejde-67 15 22 . . PUNCT ejde-67 16 1 for for ADP ejde-67 16 2 information information NOUN ejde-67 16 3 of of ADP ejde-67 16 4 the the DET ejde-67 16 5 reader reader NOUN ejde-67 16 6 , , PUNCT ejde-67 16 7 we we PRON ejde-67 16 8 refer refer VERB ejde-67 16 9 to to ADP ejde-67 16 10 n’guérékata n’guérékata NOUN ejde-67 16 11 ’s ’s PART ejde-67 16 12 book book NOUN ejde-67 16 13 [ [ X ejde-67 16 14 15 15 NUM ejde-67 16 15 ] ] PUNCT ejde-67 16 16 . . PUNCT ejde-67 17 1 in in ADP ejde-67 17 2 [ [ X ejde-67 17 3 14 14 NUM ejde-67 17 4 ] ] PUNCT ejde-67 17 5 , , PUNCT ejde-67 17 6 the the DET ejde-67 17 7 author author NOUN ejde-67 17 8 proved prove VERB ejde-67 17 9 the the DET ejde-67 17 10 existence existence NOUN ejde-67 17 11 of of ADP ejde-67 17 12 almost almost ADV ejde-67 17 13 automorphic automorphic ADJ ejde-67 17 14 solution solution NOUN ejde-67 17 15 for for ADP ejde-67 17 16 the the DET ejde-67 17 17 ordinary ordinary ADJ ejde-67 17 18 differential differential ADJ ejde-67 17 19 equation equation NOUN ejde-67 17 20 x′(t x′(t NOUN ejde-67 17 21 ) ) PUNCT ejde-67 17 22 = = SYM ejde-67 17 23 hx(t hx(t X ejde-67 17 24 ) ) PUNCT ejde-67 18 1 + + CCONJ ejde-67 18 2 e(t e(t NOUN ejde-67 18 3 ) ) PUNCT ejde-67 18 4 t t PROPN ejde-67 18 5 ∈ ∈ PROPN ejde-67 18 6 r r NOUN ejde-67 18 7 , , PUNCT ejde-67 18 8 where where SCONJ ejde-67 18 9 h h NOUN ejde-67 18 10 is be AUX ejde-67 18 11 a a DET ejde-67 18 12 constant constant ADJ ejde-67 18 13 ( ( PUNCT ejde-67 18 14 n× n× INTJ ejde-67 18 15 n)-matrix n)-matrix PUNCT ejde-67 18 16 and and CCONJ ejde-67 18 17 e e X ejde-67 18 18 : : PUNCT ejde-67 18 19 r→ r→ PROPN ejde-67 18 20 rn rn PROPN ejde-67 18 21 is be AUX ejde-67 18 22 almost almost ADV ejde-67 18 23 automorphic automorphic ADJ ejde-67 18 24 . . PUNCT ejde-67 19 1 he he PRON ejde-67 19 2 proved prove VERB ejde-67 19 3 that that SCONJ ejde-67 19 4 the the DET ejde-67 19 5 existence existence NOUN ejde-67 19 6 of of ADP ejde-67 19 7 a a DET ejde-67 19 8 bounded bound VERB ejde-67 19 9 solution solution NOUN ejde-67 19 10 on on ADP ejde-67 19 11 r+ r+ PUNCT ejde-67 19 12 implies imply VERB ejde-67 19 13 the the DET ejde-67 19 14 existence existence NOUN ejde-67 19 15 of of ADP ejde-67 19 16 an an DET ejde-67 19 17 2020 2020 NUM ejde-67 19 18 mathematics mathematic NOUN ejde-67 19 19 subject subject ADJ ejde-67 19 20 classification classification NOUN ejde-67 19 21 . . PUNCT ejde-67 20 1 35g15 35g15 NUM ejde-67 20 2 , , PUNCT ejde-67 20 3 35g20 35g20 NUM ejde-67 20 4 , , PUNCT ejde-67 20 5 35g25 35g25 NUM ejde-67 20 6 , , PUNCT ejde-67 20 7 35g30 35g30 NUM ejde-67 20 8 . . PUNCT ejde-67 21 1 key key ADJ ejde-67 21 2 words word NOUN ejde-67 21 3 and and CCONJ ejde-67 21 4 phrases phrase NOUN ejde-67 21 5 . . PUNCT ejde-67 22 1 functional functional ADJ ejde-67 22 2 differential differential NOUN ejde-67 22 3 equations equation NOUN ejde-67 22 4 ; ; PUNCT ejde-67 22 5 quasi quasi ADJ ejde-67 22 6 - - ADJ ejde-67 22 7 compact compact ADJ ejde-67 22 8 semigroup semigroup NOUN ejde-67 22 9 ; ; PUNCT ejde-67 22 10 variation variation NOUN ejde-67 22 11 of of ADP ejde-67 22 12 constants constant NOUN ejde-67 22 13 formula formula NOUN ejde-67 22 14 ; ; PUNCT ejde-67 22 15 stepanov stepanov VERB ejde-67 22 16 - - PUNCT ejde-67 22 17 almost almost ADV ejde-67 22 18 automorphic automorphic ADJ ejde-67 22 19 function function NOUN ejde-67 22 20 ; ; PUNCT ejde-67 22 21 almost almost ADV ejde-67 22 22 automorphic automorphic ADJ ejde-67 22 23 solution solution NOUN ejde-67 22 24 ; ; PUNCT ejde-67 22 25 almost almost ADV ejde-67 22 26 periodic periodic ADJ ejde-67 22 27 solution solution NOUN ejde-67 22 28 . . PUNCT ejde-67 23 1 © © ADP ejde-67 23 2 2023 2023 NUM ejde-67 23 3 . . PUNCT ejde-67 24 1 this this DET ejde-67 24 2 work work NOUN ejde-67 24 3 is be AUX ejde-67 24 4 licensed license VERB ejde-67 24 5 under under ADP ejde-67 24 6 a a DET ejde-67 24 7 cc cc NOUN ejde-67 24 8 by by ADP ejde-67 24 9 4.0 4.0 NUM ejde-67 24 10 license license NOUN ejde-67 24 11 . . PUNCT ejde-67 25 1 submitted submit VERB ejde-67 25 2 may may PROPN ejde-67 25 3 23 23 NUM ejde-67 25 4 , , PUNCT ejde-67 25 5 2022 2022 NUM ejde-67 25 6 . . PUNCT ejde-67 26 1 published publish VERB ejde-67 26 2 june june PROPN ejde-67 26 3 20 20 NUM ejde-67 26 4 , , PUNCT ejde-67 26 5 2023 2023 NUM ejde-67 26 6 . . PUNCT ejde-67 26 7 1 1 NUM ejde-67 26 8 2 2 NUM ejde-67 26 9 m. m. NOUN ejde-67 26 10 el el PROPN ejde-67 26 11 attaouy attaouy PROPN ejde-67 26 12 , , PUNCT ejde-67 26 13 k. k. PROPN ejde-67 26 14 ezzinbi ezzinbi PROPN ejde-67 26 15 , , PUNCT ejde-67 26 16 g. g. PROPN ejde-67 26 17 m. m. PROPN ejde-67 26 18 n’guérékata n’guérékata PROPN ejde-67 26 19 ejde-2023/39 ejde-2023/39 VERB ejde-67 26 20 almost almost ADV ejde-67 26 21 automorphic automorphic ADJ ejde-67 26 22 solution solution NOUN ejde-67 26 23 . . PUNCT ejde-67 27 1 in in ADP ejde-67 27 2 [ [ X ejde-67 27 3 16 16 NUM ejde-67 27 4 ] ] PUNCT ejde-67 27 5 , , PUNCT ejde-67 27 6 the the DET ejde-67 27 7 author author NOUN ejde-67 27 8 studied study VERB ejde-67 27 9 the the DET ejde-67 27 10 existence existence NOUN ejde-67 27 11 of of ADP ejde-67 27 12 almost almost ADV ejde-67 27 13 automorphic automorphic ADJ ejde-67 27 14 solutions solution NOUN ejde-67 27 15 for for ADP ejde-67 27 16 the the DET ejde-67 27 17 semilinear semilinear ADJ ejde-67 27 18 abstract abstract ADJ ejde-67 27 19 differential differential ADJ ejde-67 27 20 equation equation NOUN ejde-67 27 21 x′(t x′(t NOUN ejde-67 27 22 ) ) PUNCT ejde-67 27 23 = = SYM ejde-67 27 24 cx(t cx(t X ejde-67 27 25 ) ) PUNCT ejde-67 28 1 + + CCONJ ejde-67 28 2 θ(t θ(t PROPN ejde-67 28 3 , , PUNCT ejde-67 28 4 x(t x(t PROPN ejde-67 28 5 ) ) PUNCT ejde-67 28 6 ) ) PUNCT ejde-67 28 7 t t PROPN ejde-67 28 8 ≥ ≥ NUM ejde-67 28 9 0 0 NUM ejde-67 28 10 , , PUNCT ejde-67 28 11 ( ( PUNCT ejde-67 28 12 1.1 1.1 NUM ejde-67 28 13 ) ) PUNCT ejde-67 28 14 where where SCONJ ejde-67 28 15 c c NOUN ejde-67 28 16 generates generate VERB ejde-67 28 17 an an DET ejde-67 28 18 exponentially exponentially ADV ejde-67 28 19 stable stable ADJ ejde-67 28 20 c0 c0 NOUN ejde-67 28 21 - - PUNCT ejde-67 28 22 semigroup semigroup NOUN ejde-67 28 23 on on ADP ejde-67 28 24 a a DET ejde-67 28 25 banach banach NOUN ejde-67 28 26 space space NOUN ejde-67 28 27 e e NOUN ejde-67 28 28 and and CCONJ ejde-67 28 29 θ θ PROPN ejde-67 28 30 is be AUX ejde-67 28 31 an an DET ejde-67 28 32 almost almost ADV ejde-67 28 33 automorphic automorphic ADJ ejde-67 28 34 function function NOUN ejde-67 28 35 from from ADP ejde-67 28 36 r r NOUN ejde-67 28 37 to to ADP ejde-67 28 38 e. e. PROPN ejde-67 28 39 the the DET ejde-67 28 40 author author NOUN ejde-67 28 41 proved prove VERB ejde-67 28 42 that that SCONJ ejde-67 28 43 the the DET ejde-67 28 44 only only ADJ ejde-67 28 45 bounded bounded ADJ ejde-67 28 46 mild mild ADJ ejde-67 28 47 solution solution NOUN ejde-67 28 48 of of ADP ejde-67 28 49 ( ( PUNCT ejde-67 28 50 1.1 1.1 NUM ejde-67 28 51 ) ) PUNCT ejde-67 28 52 on on ADP ejde-67 28 53 r× r× NOUN ejde-67 28 54 e e NOUN ejde-67 28 55 is be AUX ejde-67 28 56 almost almost ADV ejde-67 28 57 automorphic automorphic ADJ ejde-67 28 58 . . PUNCT ejde-67 29 1 recently recently ADV ejde-67 29 2 ezzinbi ezzinbi NOUN ejde-67 29 3 and and CCONJ ejde-67 29 4 n’guérékata n’guérékata NOUN ejde-67 30 1 [ [ X ejde-67 30 2 11 11 NUM ejde-67 30 3 ] ] PUNCT ejde-67 30 4 established establish VERB ejde-67 30 5 the the DET ejde-67 30 6 existence existence NOUN ejde-67 30 7 of of ADP ejde-67 30 8 an an DET ejde-67 30 9 almost almost ADV ejde-67 30 10 automorphic automorphic ADJ ejde-67 30 11 solution solution NOUN ejde-67 30 12 for for ADP ejde-67 30 13 the the DET ejde-67 30 14 partial partial ADJ ejde-67 30 15 functional functional ADJ ejde-67 30 16 differential differential NOUN ejde-67 30 17 equation equation NOUN ejde-67 30 18 d d PROPN ejde-67 30 19 dt dt PUNCT ejde-67 30 20 u(t u(t PROPN ejde-67 30 21 ) ) PUNCT ejde-67 30 22 = = NOUN ejde-67 30 23 ãu(t ãu(t NOUN ejde-67 30 24 ) ) PUNCT ejde-67 31 1 + + CCONJ ejde-67 31 2 l̃(ut l̃(ut ADJ ejde-67 31 3 ) ) PUNCT ejde-67 32 1 + + CCONJ ejde-67 32 2 h̃(t h̃(t NOUN ejde-67 32 3 ) ) PUNCT ejde-67 32 4 for for ADP ejde-67 32 5 t t PROPN ejde-67 32 6 ≥ ≥ PROPN ejde-67 32 7 0 0 NUM ejde-67 32 8 , , PUNCT ejde-67 32 9 u0 u0 NOUN ejde-67 32 10 = = PUNCT ejde-67 32 11 ϕ ϕ PROPN ejde-67 32 12 ∈ ∈ PROPN ejde-67 32 13 c([−r c([−r PROPN ejde-67 32 14 , , PUNCT ejde-67 32 15 0 0 NUM ejde-67 32 16 ] ] PUNCT ejde-67 32 17 , , PUNCT ejde-67 32 18 f f PROPN ejde-67 32 19 ) ) PUNCT ejde-67 32 20 , , PUNCT ejde-67 32 21 ( ( PUNCT ejde-67 32 22 1.2 1.2 NUM ejde-67 32 23 ) ) PUNCT ejde-67 32 24 where where SCONJ ejde-67 32 25 ã ã PROPN ejde-67 32 26 is be AUX ejde-67 32 27 a a DET ejde-67 32 28 linear linear ADJ ejde-67 32 29 operator operator NOUN ejde-67 32 30 on on ADP ejde-67 32 31 a a DET ejde-67 32 32 banach banach NOUN ejde-67 32 33 space space NOUN ejde-67 32 34 f f AUX ejde-67 32 35 not not PART ejde-67 32 36 necessarily necessarily ADV ejde-67 32 37 densely densely ADV ejde-67 32 38 defined define VERB ejde-67 32 39 and and CCONJ ejde-67 32 40 satisfies satisfy VERB ejde-67 32 41 the the DET ejde-67 32 42 hille hille PROPN ejde-67 32 43 - - PUNCT ejde-67 32 44 yosida yosida PROPN ejde-67 32 45 condition condition NOUN ejde-67 32 46 , , PUNCT ejde-67 32 47 see see VERB ejde-67 32 48 [ [ X ejde-67 32 49 11 11 NUM ejde-67 32 50 ] ] PUNCT ejde-67 32 51 . . PUNCT ejde-67 33 1 l̃ l̃ PROPN ejde-67 33 2 is be AUX ejde-67 33 3 a a DET ejde-67 33 4 bounded bounded ADJ ejde-67 33 5 linear linear ADJ ejde-67 33 6 operator operator NOUN ejde-67 33 7 from from ADP ejde-67 33 8 c([−r c([−r PROPN ejde-67 33 9 , , PUNCT ejde-67 33 10 0 0 NUM ejde-67 33 11 ] ] PUNCT ejde-67 33 12 , , PUNCT ejde-67 33 13 f f PROPN ejde-67 33 14 ) ) PUNCT ejde-67 33 15 to to ADP ejde-67 33 16 f f PROPN ejde-67 33 17 with with ADP ejde-67 33 18 c([−r c([−r PROPN ejde-67 33 19 , , PUNCT ejde-67 33 20 0 0 NUM ejde-67 33 21 ] ] PUNCT ejde-67 33 22 , , PUNCT ejde-67 33 23 f f PROPN ejde-67 33 24 ) ) PUNCT ejde-67 33 25 is be AUX ejde-67 33 26 the the DET ejde-67 33 27 space space NOUN ejde-67 33 28 of of ADP ejde-67 33 29 continuous continuous ADJ ejde-67 33 30 functions function NOUN ejde-67 33 31 from from ADP ejde-67 33 32 [ [ X ejde-67 33 33 −r −r ADJ ejde-67 33 34 , , PUNCT ejde-67 33 35 0 0 NUM ejde-67 33 36 ] ] PUNCT ejde-67 33 37 to to ADP ejde-67 33 38 f f PROPN ejde-67 33 39 endowed endow VERB ejde-67 33 40 with with ADP ejde-67 33 41 the the DET ejde-67 33 42 uniform uniform ADJ ejde-67 33 43 norm norm NOUN ejde-67 33 44 topology topology NOUN ejde-67 33 45 and and CCONJ ejde-67 33 46 h̃ h̃ PROPN ejde-67 33 47 is be AUX ejde-67 33 48 an an DET ejde-67 33 49 almost almost ADV ejde-67 33 50 automorphic automorphic ADJ ejde-67 33 51 function function NOUN ejde-67 33 52 from from ADP ejde-67 33 53 r r NOUN ejde-67 33 54 to to ADP ejde-67 33 55 f f PROPN ejde-67 33 56 , , PUNCT ejde-67 33 57 the the DET ejde-67 33 58 history history NOUN ejde-67 33 59 function function VERB ejde-67 33 60 ut ut PROPN ejde-67 33 61 ∈ ∈ PROPN ejde-67 33 62 c([−r c([−r PROPN ejde-67 33 63 , , PUNCT ejde-67 33 64 0 0 NUM ejde-67 33 65 ] ] PUNCT ejde-67 33 66 , , PUNCT ejde-67 33 67 f f PROPN ejde-67 33 68 ) ) PUNCT ejde-67 33 69 is be AUX ejde-67 33 70 defined define VERB ejde-67 33 71 by by ADP ejde-67 33 72 ut(θ ut(θ ADV ejde-67 33 73 ) ) PUNCT ejde-67 33 74 = = PUNCT ejde-67 34 1 u(t+ u(t+ NOUN ejde-67 34 2 θ θ PROPN ejde-67 34 3 ) ) PUNCT ejde-67 34 4 , , PUNCT ejde-67 34 5 for for ADP ejde-67 34 6 θ θ PROPN ejde-67 34 7 ∈ ∈ PROPN ejde-67 35 1 [ [ X ejde-67 35 2 −r −r ADJ ejde-67 35 3 , , PUNCT ejde-67 35 4 0 0 NUM ejde-67 35 5 ] ] PUNCT ejde-67 35 6 . . PUNCT ejde-67 36 1 by by ADP ejde-67 36 2 developing develop VERB ejde-67 36 3 new new ADJ ejde-67 36 4 fundamental fundamental ADJ ejde-67 36 5 results result NOUN ejde-67 36 6 about about ADP ejde-67 36 7 the the DET ejde-67 36 8 spectral spectral ADJ ejde-67 36 9 analysis analysis NOUN ejde-67 36 10 of of ADP ejde-67 36 11 the the DET ejde-67 36 12 solutions solution NOUN ejde-67 36 13 and and CCONJ ejde-67 36 14 a a DET ejde-67 36 15 new new ADJ ejde-67 36 16 reduction reduction NOUN ejde-67 36 17 principle principle NOUN ejde-67 36 18 , , PUNCT ejde-67 36 19 the the DET ejde-67 36 20 authors author NOUN ejde-67 36 21 proved prove VERB ejde-67 36 22 that that SCONJ ejde-67 36 23 the the DET ejde-67 36 24 existence existence NOUN ejde-67 36 25 of of ADP ejde-67 36 26 a a DET ejde-67 36 27 bounded bound VERB ejde-67 36 28 solution solution NOUN ejde-67 36 29 on on ADP ejde-67 36 30 r+ r+ NOUN ejde-67 36 31 of of ADP ejde-67 36 32 ( ( PUNCT ejde-67 36 33 1.2 1.2 NUM ejde-67 36 34 ) ) PUNCT ejde-67 36 35 is be AUX ejde-67 36 36 equivalent equivalent ADJ ejde-67 36 37 to to ADP ejde-67 36 38 the the DET ejde-67 36 39 existence existence NOUN ejde-67 36 40 of of ADP ejde-67 36 41 an an DET ejde-67 36 42 almost almost ADV ejde-67 36 43 automorphic automorphic ADJ ejde-67 36 44 solution solution NOUN ejde-67 36 45 . . PUNCT ejde-67 37 1 in in ADP ejde-67 37 2 this this DET ejde-67 37 3 work work NOUN ejde-67 37 4 we we PRON ejde-67 37 5 are be AUX ejde-67 37 6 interested interested ADJ ejde-67 37 7 in in ADP ejde-67 37 8 investigating investigate VERB ejde-67 37 9 the the DET ejde-67 37 10 existence existence NOUN ejde-67 37 11 of of ADP ejde-67 37 12 almost almost ADV ejde-67 37 13 automorphic automorphic ADJ ejde-67 37 14 and and CCONJ ejde-67 37 15 almost almost ADV ejde-67 37 16 periodic periodic ADJ ejde-67 37 17 solutions solution NOUN ejde-67 37 18 for for ADP ejde-67 37 19 the the DET ejde-67 37 20 partial partial ADJ ejde-67 37 21 functional functional ADJ ejde-67 37 22 differential differential NOUN ejde-67 37 23 equation equation NOUN ejde-67 37 24 x′(t x′(t NOUN ejde-67 37 25 ) ) PUNCT ejde-67 37 26 = = SYM ejde-67 37 27 ax(t ax(t NUM ejde-67 37 28 ) ) PUNCT ejde-67 38 1 + + CCONJ ejde-67 38 2 l(xt l(xt NOUN ejde-67 38 3 ) ) PUNCT ejde-67 38 4 + + CCONJ ejde-67 38 5 f(t f(t NOUN ejde-67 38 6 ) ) PUNCT ejde-67 38 7 for for ADP ejde-67 38 8 t t PROPN ejde-67 38 9 ∈ ∈ PROPN ejde-67 38 10 r r PROPN ejde-67 38 11 , , PUNCT ejde-67 38 12 ( ( PUNCT ejde-67 38 13 1.3 1.3 NUM ejde-67 38 14 ) ) PUNCT ejde-67 38 15 wherea wherea NOUN ejde-67 38 16 is be AUX ejde-67 38 17 the the DET ejde-67 38 18 infinitesimal infinitesimal ADJ ejde-67 38 19 generator generator NOUN ejde-67 38 20 of of ADP ejde-67 38 21 a a DET ejde-67 38 22 strongly strongly ADV ejde-67 38 23 continuous continuous ADJ ejde-67 38 24 semigroup semigroup NOUN ejde-67 38 25 of of ADP ejde-67 38 26 bounded bounded PROPN ejde-67 38 27 linear linear PROPN ejde-67 38 28 operators operators PROPN ejde-67 38 29 t t PROPN ejde-67 38 30 ( ( PUNCT ejde-67 38 31 t t PROPN ejde-67 38 32 ) ) PUNCT ejde-67 38 33 on on ADP ejde-67 38 34 a a DET ejde-67 38 35 banach banach NOUN ejde-67 38 36 space space NOUN ejde-67 38 37 x. x. NOUN ejde-67 38 38 x(t x(t PROPN ejde-67 38 39 ) ) PUNCT ejde-67 38 40 ∈ ∈ PROPN ejde-67 38 41 x x X ejde-67 38 42 , , PUNCT ejde-67 38 43 l l NOUN ejde-67 38 44 is be AUX ejde-67 38 45 a a DET ejde-67 38 46 bounded bounded ADJ ejde-67 38 47 linear linear ADJ ejde-67 38 48 operator operator NOUN ejde-67 38 49 from from ADP ejde-67 38 50 c([−r c([−r PROPN ejde-67 38 51 , , PUNCT ejde-67 38 52 0 0 NUM ejde-67 38 53 ] ] PUNCT ejde-67 38 54 , , PUNCT ejde-67 38 55 x x X ejde-67 38 56 ) ) PUNCT ejde-67 38 57 to to ADP ejde-67 38 58 x x PUNCT ejde-67 38 59 with with ADP ejde-67 38 60 c([−r c([−r PROPN ejde-67 38 61 , , PUNCT ejde-67 38 62 0 0 NUM ejde-67 38 63 ] ] PUNCT ejde-67 38 64 , , PUNCT ejde-67 38 65 x x X ejde-67 38 66 ) ) PUNCT ejde-67 38 67 is be AUX ejde-67 38 68 the the DET ejde-67 38 69 space space NOUN ejde-67 38 70 of of ADP ejde-67 38 71 continuous continuous ADJ ejde-67 38 72 functions function NOUN ejde-67 38 73 from from ADP ejde-67 38 74 [ [ X ejde-67 38 75 −r −r ADJ ejde-67 38 76 , , PUNCT ejde-67 38 77 0 0 NUM ejde-67 38 78 ] ] PUNCT ejde-67 38 79 to to PART ejde-67 38 80 x x SYM ejde-67 38 81 endowed endow VERB ejde-67 38 82 with with ADP ejde-67 38 83 the the DET ejde-67 38 84 uniform uniform ADJ ejde-67 38 85 norm norm NOUN ejde-67 38 86 topology topology NOUN ejde-67 38 87 and and CCONJ ejde-67 38 88 r r NOUN ejde-67 38 89 > > X ejde-67 38 90 0 0 NUM ejde-67 38 91 . . PUNCT ejde-67 39 1 the the DET ejde-67 39 2 history history NOUN ejde-67 39 3 function function NOUN ejde-67 39 4 xt xt ADP ejde-67 39 5 ∈ ∈ PROPN ejde-67 39 6 c([−r c([−r PROPN ejde-67 39 7 , , PUNCT ejde-67 39 8 0 0 NUM ejde-67 39 9 ] ] PUNCT ejde-67 39 10 , , PUNCT ejde-67 39 11 x x X ejde-67 39 12 ) ) PUNCT ejde-67 39 13 is be AUX ejde-67 39 14 defined define VERB ejde-67 39 15 by by ADP ejde-67 39 16 xt xt PROPN ejde-67 39 17 : : PUNCT ejde-67 40 1 [ [ X ejde-67 40 2 −r −r ADJ ejde-67 40 3 , , PUNCT ejde-67 40 4 0]→ 0]→ NOUN ejde-67 40 5 x x NOUN ejde-67 40 6 , , PUNCT ejde-67 40 7 xt(θ xt(θ NUM ejde-67 40 8 ) ) PUNCT ejde-67 40 9 = = SYM ejde-67 41 1 x(t+ x(t+ PROPN ejde-67 41 2 θ θ X ejde-67 41 3 ) ) PUNCT ejde-67 41 4 . . PUNCT ejde-67 42 1 the the DET ejde-67 42 2 function function NOUN ejde-67 42 3 f f NOUN ejde-67 42 4 : : PUNCT ejde-67 42 5 r r X ejde-67 42 6 → → PUNCT ejde-67 42 7 x x X ejde-67 42 8 is be AUX ejde-67 42 9 a a DET ejde-67 42 10 stepanov stepanov ADJ ejde-67 42 11 almost almost ADV ejde-67 42 12 automorphic automorphic ADJ ejde-67 42 13 function function NOUN ejde-67 42 14 , , PUNCT ejde-67 42 15 which which PRON ejde-67 42 16 is be AUX ejde-67 42 17 a a DET ejde-67 42 18 weaker weak ADJ ejde-67 42 19 notion notion NOUN ejde-67 42 20 of of ADP ejde-67 42 21 almost almost ADV ejde-67 42 22 automorphy automorphy NOUN ejde-67 42 23 . . PUNCT ejde-67 43 1 for for ADP ejde-67 43 2 more more ADJ ejde-67 43 3 details detail NOUN ejde-67 43 4 on on ADP ejde-67 43 5 stepanov stepanov VERB ejde-67 43 6 almost almost ADV ejde-67 43 7 automorphic automorphic ADJ ejde-67 43 8 functions function NOUN ejde-67 43 9 , , PUNCT ejde-67 43 10 we we PRON ejde-67 43 11 refer refer VERB ejde-67 43 12 the the DET ejde-67 43 13 reader reader NOUN ejde-67 43 14 to to ADP ejde-67 43 15 [ [ X ejde-67 43 16 17 17 NUM ejde-67 43 17 ] ] PUNCT ejde-67 43 18 . . PUNCT ejde-67 44 1 the the DET ejde-67 44 2 usual usual ADJ ejde-67 44 3 condition condition NOUN ejde-67 44 4 for for ADP ejde-67 44 5 studying study VERB ejde-67 44 6 the the DET ejde-67 44 7 existence existence NOUN ejde-67 44 8 of of ADP ejde-67 44 9 almost almost ADV ejde-67 44 10 periodic periodic ADJ ejde-67 44 11 and and CCONJ ejde-67 44 12 almost almost ADV ejde-67 44 13 automorphic automorphic ADJ ejde-67 44 14 solutions solution NOUN ejde-67 44 15 for for ADP ejde-67 44 16 this this DET ejde-67 44 17 problem problem NOUN ejde-67 44 18 is be AUX ejde-67 44 19 that that SCONJ ejde-67 44 20 the the DET ejde-67 44 21 c0 c0 NOUN ejde-67 44 22 - - PUNCT ejde-67 44 23 semigroup semigroup PROPN ejde-67 44 24 ( ( PUNCT ejde-67 44 25 t t PROPN ejde-67 44 26 ( ( PUNCT ejde-67 44 27 t))t≥0 t))t≥0 PROPN ejde-67 44 28 is be AUX ejde-67 44 29 compact compact ADJ ejde-67 44 30 . . PUNCT ejde-67 45 1 for for ADP ejde-67 45 2 example example NOUN ejde-67 45 3 , , PUNCT ejde-67 45 4 the the DET ejde-67 45 5 problem problem NOUN ejde-67 45 6 of of ADP ejde-67 45 7 existence existence NOUN ejde-67 45 8 of of ADP ejde-67 45 9 almost almost ADV ejde-67 45 10 automorphic automorphic ADJ ejde-67 45 11 solutions solution NOUN ejde-67 45 12 has have AUX ejde-67 45 13 been be AUX ejde-67 45 14 studied study VERB ejde-67 45 15 recently recently ADV ejde-67 45 16 by by ADP ejde-67 45 17 benkhalti benkhalti NOUN ejde-67 45 18 , , PUNCT ejde-67 45 19 es es NOUN ejde-67 45 20 - - PUNCT ejde-67 45 21 sebbar sebbar NOUN ejde-67 45 22 and and CCONJ ejde-67 45 23 ezzinbi ezzinbi NOUN ejde-67 45 24 in in ADP ejde-67 45 25 [ [ X ejde-67 45 26 2 2 NUM ejde-67 45 27 ] ] PUNCT ejde-67 45 28 , , PUNCT ejde-67 45 29 using use VERB ejde-67 45 30 this this DET ejde-67 45 31 condition condition NOUN ejde-67 45 32 . . PUNCT ejde-67 46 1 our our PRON ejde-67 46 2 aim aim NOUN ejde-67 46 3 in in ADP ejde-67 46 4 this this DET ejde-67 46 5 paper paper NOUN ejde-67 46 6 is be AUX ejde-67 46 7 to to PART ejde-67 46 8 establish establish VERB ejde-67 46 9 the the DET ejde-67 46 10 existence existence NOUN ejde-67 46 11 of of ADP ejde-67 46 12 almost almost ADV ejde-67 46 13 automorphic automorphic ADJ ejde-67 46 14 and and CCONJ ejde-67 46 15 almost almost ADV ejde-67 46 16 periodic periodic ADJ ejde-67 46 17 solutions solution NOUN ejde-67 46 18 for for ADP ejde-67 46 19 a a DET ejde-67 46 20 class class NOUN ejde-67 46 21 of of ADP ejde-67 46 22 equations equation NOUN ejde-67 46 23 in in ADP ejde-67 46 24 which which PRON ejde-67 46 25 the the DET ejde-67 46 26 c0 c0 NOUN ejde-67 46 27 - - PUNCT ejde-67 46 28 semigroup semigroup PROPN ejde-67 46 29 ( ( PUNCT ejde-67 46 30 t t PROPN ejde-67 46 31 ( ( PUNCT ejde-67 46 32 t))t≥0 t))t≥0 PROPN ejde-67 46 33 is be AUX ejde-67 46 34 not not PART ejde-67 46 35 necessarily necessarily ADV ejde-67 46 36 compact compact ADJ ejde-67 46 37 . . PUNCT ejde-67 47 1 in in ADP ejde-67 47 2 the the DET ejde-67 47 3 last last ADJ ejde-67 47 4 direction direction NOUN ejde-67 47 5 , , PUNCT ejde-67 47 6 we we PRON ejde-67 47 7 refer refer VERB ejde-67 47 8 the the DET ejde-67 47 9 reader reader NOUN ejde-67 47 10 to to ADP ejde-67 47 11 [ [ X ejde-67 47 12 12 12 NUM ejde-67 47 13 ] ] PUNCT ejde-67 47 14 , , PUNCT ejde-67 47 15 where where SCONJ ejde-67 47 16 the the DET ejde-67 47 17 authors author NOUN ejde-67 47 18 studied study VERB ejde-67 47 19 the the DET ejde-67 47 20 existence existence NOUN ejde-67 47 21 of of ADP ejde-67 47 22 almost almost ADV ejde-67 47 23 periodic periodic ADJ ejde-67 47 24 solutions solution NOUN ejde-67 47 25 of of ADP ejde-67 47 26 ( ( PUNCT ejde-67 47 27 1.3 1.3 NUM ejde-67 47 28 ) ) PUNCT ejde-67 47 29 when when SCONJ ejde-67 47 30 the the DET ejde-67 47 31 c0 c0 NOUN ejde-67 47 32 - - PUNCT ejde-67 47 33 semigroup semigroup PROPN ejde-67 47 34 ( ( PUNCT ejde-67 47 35 t t PROPN ejde-67 47 36 ( ( PUNCT ejde-67 47 37 t))t≥0 t))t≥0 PROPN ejde-67 47 38 is be AUX ejde-67 47 39 not not PART ejde-67 47 40 compact compact ADJ ejde-67 47 41 but but CCONJ ejde-67 47 42 the the DET ejde-67 47 43 operator operator NOUN ejde-67 47 44 t t PROPN ejde-67 47 45 ( ( PUNCT ejde-67 47 46 t)l t)l NOUN ejde-67 47 47 is be AUX ejde-67 47 48 compact compact ADJ ejde-67 47 49 for for ADP ejde-67 47 50 t t PROPN ejde-67 47 51 > > X ejde-67 47 52 0 0 PUNCT ejde-67 48 1 and and CCONJ ejde-67 48 2 the the DET ejde-67 48 3 input input NOUN ejde-67 48 4 term term NOUN ejde-67 48 5 f f PROPN ejde-67 48 6 is be AUX ejde-67 48 7 almost almost ADV ejde-67 48 8 periodic periodic ADJ ejde-67 48 9 . . PUNCT ejde-67 49 1 as as ADP ejde-67 49 2 an an DET ejde-67 49 3 extension extension NOUN ejde-67 49 4 of of ADP ejde-67 49 5 the the DET ejde-67 49 6 work work NOUN ejde-67 49 7 [ [ X ejde-67 49 8 12 12 NUM ejde-67 49 9 ] ] PUNCT ejde-67 49 10 , , PUNCT ejde-67 49 11 we we PRON ejde-67 49 12 prove prove VERB ejde-67 49 13 that that SCONJ ejde-67 49 14 the the DET ejde-67 49 15 ( ( PUNCT ejde-67 49 16 1.3 1.3 NUM ejde-67 49 17 ) ) PUNCT ejde-67 49 18 has have VERB ejde-67 49 19 an an DET ejde-67 49 20 almost almost ADV ejde-67 49 21 periodic periodic ADJ ejde-67 49 22 solution solution NOUN ejde-67 49 23 under under ADP ejde-67 49 24 the the DET ejde-67 49 25 hypothesis hypothesis NOUN ejde-67 49 26 that that PRON ejde-67 49 27 the the DET ejde-67 49 28 operator operator NOUN ejde-67 49 29 t t PROPN ejde-67 49 30 ( ( PUNCT ejde-67 49 31 t)l t)l NOUN ejde-67 49 32 is be AUX ejde-67 49 33 compact compact ADJ ejde-67 49 34 for for ADP ejde-67 49 35 t t PROPN ejde-67 49 36 > > X ejde-67 49 37 0 0 PUNCT ejde-67 50 1 and and CCONJ ejde-67 50 2 the the DET ejde-67 50 3 input input NOUN ejde-67 50 4 term term NOUN ejde-67 50 5 f f PROPN ejde-67 50 6 is be AUX ejde-67 50 7 only only ADV ejde-67 50 8 stepanov stepanov VERB ejde-67 50 9 almost almost ADV ejde-67 50 10 periodic periodic ADJ ejde-67 50 11 . . PUNCT ejde-67 51 1 for for ADP ejde-67 51 2 more more ADJ ejde-67 51 3 details detail NOUN ejde-67 51 4 on on ADP ejde-67 51 5 stepanov stepanov ADJ ejde-67 51 6 ejde-2023/39 ejde-2023/39 ADJ ejde-67 51 7 reduction reduction NOUN ejde-67 51 8 principle principle NOUN ejde-67 51 9 3 3 NUM ejde-67 52 1 almost almost ADV ejde-67 52 2 periodic periodic ADJ ejde-67 52 3 functions function NOUN ejde-67 52 4 , , PUNCT ejde-67 52 5 we we PRON ejde-67 52 6 refer refer VERB ejde-67 52 7 the the DET ejde-67 52 8 reader reader NOUN ejde-67 52 9 to to ADP ejde-67 52 10 [ [ X ejde-67 52 11 8 8 NUM ejde-67 52 12 ] ] PUNCT ejde-67 52 13 . . PUNCT ejde-67 53 1 also also ADV ejde-67 53 2 extend extend VERB ejde-67 53 3 our our PRON ejde-67 53 4 results result NOUN ejde-67 53 5 to to ADP ejde-67 53 6 the the DET ejde-67 53 7 almost almost ADV ejde-67 53 8 automorphic automorphic ADJ ejde-67 53 9 case case NOUN ejde-67 53 10 . . PUNCT ejde-67 54 1 we we PRON ejde-67 54 2 prove prove VERB ejde-67 54 3 that that SCONJ ejde-67 54 4 the the DET ejde-67 54 5 ( ( PUNCT ejde-67 54 6 1.3 1.3 NUM ejde-67 54 7 ) ) PUNCT ejde-67 54 8 has have VERB ejde-67 54 9 an an DET ejde-67 54 10 almost almost ADV ejde-67 54 11 automorphic automorphic ADJ ejde-67 54 12 solution solution NOUN ejde-67 54 13 if if SCONJ ejde-67 54 14 the the DET ejde-67 54 15 function function NOUN ejde-67 54 16 f f PROPN ejde-67 54 17 is be AUX ejde-67 54 18 just just ADV ejde-67 54 19 stepanov stepanov VERB ejde-67 54 20 almost almost ADV ejde-67 54 21 automorphic automorphic ADJ ejde-67 54 22 . . PUNCT ejde-67 55 1 to to PART ejde-67 55 2 achieve achieve VERB ejde-67 55 3 this this DET ejde-67 55 4 goal goal NOUN ejde-67 55 5 , , PUNCT ejde-67 55 6 we we PRON ejde-67 55 7 use use VERB ejde-67 55 8 the the DET ejde-67 55 9 formula formula NOUN ejde-67 55 10 for for ADP ejde-67 55 11 the the DET ejde-67 55 12 variation variation NOUN ejde-67 55 13 of of ADP ejde-67 55 14 constants constant NOUN ejde-67 55 15 and and CCONJ ejde-67 55 16 the the DET ejde-67 55 17 reduction reduction NOUN ejde-67 55 18 principle principle NOUN ejde-67 55 19 developed develop VERB ejde-67 55 20 by by ADP ejde-67 55 21 ezzinbi ezzinbi NOUN ejde-67 55 22 and and CCONJ ejde-67 55 23 n’guérékata n’guérékata NOUN ejde-67 55 24 in in ADP ejde-67 55 25 [ [ X ejde-67 55 26 11 11 NUM ejde-67 55 27 ] ] PUNCT ejde-67 55 28 . . PUNCT ejde-67 56 1 this this DET ejde-67 56 2 work work NOUN ejde-67 56 3 is be AUX ejde-67 56 4 organized organize VERB ejde-67 56 5 as as SCONJ ejde-67 56 6 follows follow VERB ejde-67 56 7 : : PUNCT ejde-67 56 8 in in ADP ejde-67 56 9 section section NOUN ejde-67 56 10 2 2 NUM ejde-67 56 11 , , PUNCT ejde-67 56 12 we we PRON ejde-67 56 13 recall recall VERB ejde-67 56 14 some some DET ejde-67 56 15 results result NOUN ejde-67 56 16 on on ADP ejde-67 56 17 partial partial ADJ ejde-67 56 18 functional functional ADJ ejde-67 56 19 differential differential NOUN ejde-67 56 20 equations equation NOUN ejde-67 56 21 and and CCONJ ejde-67 56 22 we we PRON ejde-67 56 23 establish establish VERB ejde-67 56 24 fundamental fundamental ADJ ejde-67 56 25 results result NOUN ejde-67 56 26 about about ADP ejde-67 56 27 the the DET ejde-67 56 28 spectral spectral ADJ ejde-67 56 29 decomposition decomposition NOUN ejde-67 56 30 of of ADP ejde-67 56 31 solutions solution NOUN ejde-67 56 32 of of ADP ejde-67 56 33 ( ( PUNCT ejde-67 56 34 1.3 1.3 NUM ejde-67 56 35 ) ) PUNCT ejde-67 56 36 . . PUNCT ejde-67 57 1 in in ADP ejde-67 57 2 section section NOUN ejde-67 57 3 3 3 NUM ejde-67 57 4 , , PUNCT ejde-67 57 5 we we PRON ejde-67 57 6 develop develop VERB ejde-67 57 7 a a DET ejde-67 57 8 new new ADJ ejde-67 57 9 fundamental fundamental ADJ ejde-67 57 10 reduction reduction NOUN ejde-67 57 11 principle principle NOUN ejde-67 57 12 . . PUNCT ejde-67 58 1 section section NOUN ejde-67 58 2 4 4 NUM ejde-67 58 3 is be AUX ejde-67 58 4 devoted devote VERB ejde-67 58 5 to to ADP ejde-67 58 6 almost almost ADV ejde-67 58 7 automorphic automorphic ADJ ejde-67 58 8 and and CCONJ ejde-67 58 9 almost almost ADV ejde-67 58 10 periodic periodic ADJ ejde-67 58 11 functions function NOUN ejde-67 58 12 in in ADP ejde-67 58 13 both both CCONJ ejde-67 58 14 the the DET ejde-67 58 15 bochner bochner NOUN ejde-67 58 16 and and CCONJ ejde-67 58 17 the the DET ejde-67 58 18 stepanov stepanov ADJ ejde-67 58 19 senses sense NOUN ejde-67 58 20 . . PUNCT ejde-67 59 1 in in ADP ejde-67 59 2 section section NOUN ejde-67 59 3 5 5 NUM ejde-67 59 4 , , PUNCT ejde-67 59 5 we we PRON ejde-67 59 6 study study VERB ejde-67 59 7 the the DET ejde-67 59 8 existence existence NOUN ejde-67 59 9 of of ADP ejde-67 59 10 almost almost ADV ejde-67 59 11 automorphic automorphic ADJ ejde-67 59 12 and and CCONJ ejde-67 59 13 almost almost ADV ejde-67 59 14 periodic periodic ADJ ejde-67 59 15 solutions solution NOUN ejde-67 59 16 of of ADP ejde-67 59 17 ( ( PUNCT ejde-67 59 18 1.3 1.3 NUM ejde-67 59 19 ) ) PUNCT ejde-67 59 20 through through ADP ejde-67 59 21 a a DET ejde-67 59 22 new new ADJ ejde-67 59 23 reduction reduction NOUN ejde-67 59 24 principle principle NOUN ejde-67 59 25 . . PUNCT ejde-67 60 1 in in ADP ejde-67 60 2 section section NOUN ejde-67 60 3 6 6 NUM ejde-67 60 4 , , PUNCT ejde-67 60 5 we we PRON ejde-67 60 6 illustrate illustrate VERB ejde-67 60 7 our our PRON ejde-67 60 8 results result NOUN ejde-67 60 9 to to ADP ejde-67 60 10 the the DET ejde-67 60 11 transportation transportation NOUN ejde-67 60 12 equation equation NOUN ejde-67 60 13 . . PUNCT ejde-67 61 1 the the DET ejde-67 61 2 last last ADJ ejde-67 61 3 one one NOUN ejde-67 61 4 is be AUX ejde-67 61 5 a a DET ejde-67 61 6 conclusion conclusion NOUN ejde-67 61 7 . . PUNCT ejde-67 62 1 2 2 X ejde-67 62 2 . . X ejde-67 62 3 variation variation NOUN ejde-67 62 4 of of ADP ejde-67 62 5 constants constant NOUN ejde-67 62 6 formula formula NOUN ejde-67 62 7 and and CCONJ ejde-67 62 8 spectral spectral ADJ ejde-67 62 9 decomposition decomposition NOUN ejde-67 62 10 throughout throughout ADP ejde-67 62 11 this this DET ejde-67 62 12 work work NOUN ejde-67 62 13 , , PUNCT ejde-67 62 14 ( ( PUNCT ejde-67 62 15 x x X ejde-67 62 16 , , PUNCT ejde-67 62 17 ‖ ‖ PROPN ejde-67 62 18 · · PUNCT ejde-67 62 19 ‖ ‖ NUM ejde-67 62 20 ) ) PUNCT ejde-67 62 21 is be AUX ejde-67 62 22 a a DET ejde-67 62 23 banach banach NOUN ejde-67 62 24 space space NOUN ejde-67 62 25 and and CCONJ ejde-67 62 26 c c NOUN ejde-67 62 27 = = SYM ejde-67 62 28 c([−r c([−r PROPN ejde-67 62 29 , , PUNCT ejde-67 62 30 0 0 NUM ejde-67 62 31 ] ] PUNCT ejde-67 62 32 , , PUNCT ejde-67 62 33 x x X ejde-67 62 34 ) ) PUNCT ejde-67 62 35 is be AUX ejde-67 62 36 the the DET ejde-67 62 37 space space NOUN ejde-67 62 38 of of ADP ejde-67 62 39 all all DET ejde-67 62 40 continuous continuous ADJ ejde-67 62 41 functions function NOUN ejde-67 62 42 from from ADP ejde-67 62 43 [ [ X ejde-67 62 44 −r −r ADJ ejde-67 62 45 , , PUNCT ejde-67 62 46 0 0 NUM ejde-67 62 47 ] ] PUNCT ejde-67 62 48 to to PART ejde-67 62 49 x x SYM ejde-67 62 50 endowed endow VERB ejde-67 62 51 with with ADP ejde-67 62 52 the the DET ejde-67 62 53 uniform uniform ADJ ejde-67 62 54 norm norm PROPN ejde-67 62 55 topology topology NOUN ejde-67 62 56 . . PUNCT ejde-67 63 1 let let VERB ejde-67 63 2 l(x l(x PROPN ejde-67 63 3 ) ) PUNCT ejde-67 63 4 be be AUX ejde-67 63 5 the the DET ejde-67 63 6 space space NOUN ejde-67 63 7 of of ADP ejde-67 63 8 linear linear PROPN ejde-67 63 9 and and CCONJ ejde-67 63 10 bounded bound VERB ejde-67 63 11 maps map NOUN ejde-67 63 12 from from ADP ejde-67 63 13 x x PUNCT ejde-67 63 14 to to ADP ejde-67 63 15 x x PROPN ejde-67 63 16 and and CCONJ ejde-67 63 17 k(x k(x PROPN ejde-67 63 18 ) ) PUNCT ejde-67 63 19 be be VERB ejde-67 63 20 the the DET ejde-67 63 21 space space NOUN ejde-67 63 22 of of ADP ejde-67 63 23 all all DET ejde-67 63 24 compact compact ADJ ejde-67 63 25 operators operator NOUN ejde-67 63 26 on on ADP ejde-67 63 27 x. x. NOUN ejde-67 64 1 we we PRON ejde-67 64 2 assume assume VERB ejde-67 64 3 that that SCONJ ejde-67 64 4 the the DET ejde-67 64 5 operator operator NOUN ejde-67 64 6 a a DET ejde-67 64 7 satisfies satisfie NOUN ejde-67 64 8 the the DET ejde-67 64 9 following follow VERB ejde-67 64 10 condition condition NOUN ejde-67 64 11 : : PUNCT ejde-67 64 12 ( ( PUNCT ejde-67 64 13 h1 h1 PROPN ejde-67 64 14 ) ) PUNCT ejde-67 64 15 a a PRON ejde-67 64 16 is be AUX ejde-67 64 17 the the DET ejde-67 64 18 infinitesimal infinitesimal ADJ ejde-67 64 19 generator generator NOUN ejde-67 64 20 of of ADP ejde-67 64 21 a a DET ejde-67 64 22 c0 c0 NOUN ejde-67 64 23 - - PUNCT ejde-67 64 24 semigroup semigroup NOUN ejde-67 64 25 of of ADP ejde-67 64 26 bounded bounded ADJ ejde-67 64 27 linear linear PROPN ejde-67 64 28 operators operator NOUN ejde-67 64 29 ( ( PUNCT ejde-67 64 30 t t PROPN ejde-67 64 31 ( ( PUNCT ejde-67 64 32 t))t≥0 t))t≥0 VERB ejde-67 64 33 on on ADP ejde-67 64 34 a a DET ejde-67 64 35 banach banach NOUN ejde-67 64 36 space space NOUN ejde-67 64 37 x. x. NOUN ejde-67 65 1 l l NOUN ejde-67 65 2 : : PUNCT ejde-67 66 1 c c X ejde-67 66 2 → → PUNCT ejde-67 66 3 x x X ejde-67 66 4 is be AUX ejde-67 66 5 a a DET ejde-67 66 6 bounded bounded ADJ ejde-67 66 7 linear linear ADJ ejde-67 66 8 operator operator NOUN ejde-67 66 9 on on ADP ejde-67 66 10 c c PROPN ejde-67 66 11 and and CCONJ ejde-67 66 12 f f X ejde-67 66 13 : : PUNCT ejde-67 67 1 r→ r→ PROPN ejde-67 67 2 x x X ejde-67 67 3 is be AUX ejde-67 67 4 a a DET ejde-67 67 5 continuous continuous ADJ ejde-67 67 6 function function NOUN ejde-67 67 7 from from ADP ejde-67 67 8 r r NOUN ejde-67 67 9 to to PART ejde-67 67 10 x. x. NOUN ejde-67 67 11 to to ADP ejde-67 67 12 ( ( PUNCT ejde-67 67 13 1.3 1.3 NUM ejde-67 67 14 ) ) PUNCT ejde-67 67 15 , , PUNCT ejde-67 67 16 we we PRON ejde-67 67 17 associate associate VERB ejde-67 67 18 the the DET ejde-67 67 19 problem problem NOUN ejde-67 67 20 x′(t x′(t NOUN ejde-67 67 21 ) ) PUNCT ejde-67 67 22 = = SYM ejde-67 67 23 ax(t ax(t NUM ejde-67 67 24 ) ) PUNCT ejde-67 68 1 + + CCONJ ejde-67 68 2 l(xt l(xt NOUN ejde-67 68 3 ) ) PUNCT ejde-67 68 4 + + CCONJ ejde-67 68 5 f(t f(t NOUN ejde-67 68 6 ) ) PUNCT ejde-67 68 7 for for ADP ejde-67 68 8 t t PROPN ejde-67 68 9 ≥ ≥ PROPN ejde-67 68 10 σ σ PROPN ejde-67 68 11 , , PUNCT ejde-67 68 12 xσ xσ PUNCT ejde-67 68 13 = = SYM ejde-67 68 14 ϕ ϕ PROPN ejde-67 68 15 ∈ ∈ PROPN ejde-67 68 16 c c PROPN ejde-67 68 17 , , PUNCT ejde-67 68 18 ( ( PUNCT ejde-67 68 19 2.1 2.1 NUM ejde-67 68 20 ) ) PUNCT ejde-67 68 21 we we PRON ejde-67 68 22 refer refer VERB ejde-67 68 23 to to ADP ejde-67 68 24 engel engel PROPN ejde-67 68 25 and and CCONJ ejde-67 68 26 nagel nagel PROPN ejde-67 69 1 [ [ X ejde-67 69 2 10 10 NUM ejde-67 69 3 ] ] PUNCT ejde-67 69 4 , , PUNCT ejde-67 69 5 and and CCONJ ejde-67 69 6 to to ADP ejde-67 69 7 wu wu PROPN ejde-67 70 1 [ [ X ejde-67 70 2 19 19 NUM ejde-67 70 3 ] ] PUNCT ejde-67 70 4 for for ADP ejde-67 70 5 the the DET ejde-67 70 6 basic basic ADJ ejde-67 70 7 properties property NOUN ejde-67 70 8 of of ADP ejde-67 70 9 the the DET ejde-67 70 10 problem problem NOUN ejde-67 70 11 ( ( PUNCT ejde-67 70 12 2.1 2.1 NUM ejde-67 70 13 ) ) PUNCT ejde-67 70 14 . . PUNCT ejde-67 71 1 we we PRON ejde-67 71 2 only only ADV ejde-67 71 3 mention mention VERB ejde-67 71 4 here here ADV ejde-67 71 5 that that PRON ejde-67 71 6 ( ( PUNCT ejde-67 71 7 2.1 2.1 NUM ejde-67 71 8 ) ) PUNCT ejde-67 71 9 with with ADP ejde-67 71 10 the the DET ejde-67 71 11 initial initial ADJ ejde-67 71 12 condition condition NOUN ejde-67 71 13 xσ xσ ADP ejde-67 72 1 = = SYM ejde-67 72 2 ϕ ϕ PROPN ejde-67 72 3 , , PUNCT ejde-67 72 4 has have VERB ejde-67 72 5 a a DET ejde-67 72 6 unique unique ADJ ejde-67 72 7 mild mild ADJ ejde-67 72 8 solution solution NOUN ejde-67 72 9 x x X ejde-67 72 10 ( ( PUNCT ejde-67 72 11 · · PUNCT ejde-67 72 12 , , PUNCT ejde-67 72 13 σ σ PROPN ejde-67 72 14 , , PUNCT ejde-67 72 15 ϕ ϕ PROPN ejde-67 72 16 , , PUNCT ejde-67 72 17 f f NOUN ejde-67 72 18 ) ) PUNCT ejde-67 72 19 . . PUNCT ejde-67 73 1 this this PRON ejde-67 73 2 signifies signify VERB ejde-67 73 3 that that SCONJ ejde-67 73 4 x x X ejde-67 73 5 : : PUNCT ejde-67 74 1 [ [ X ejde-67 74 2 σ σ X ejde-67 74 3 − − PROPN ejde-67 74 4 r,∞ r,∞ NOUN ejde-67 74 5 ) ) PUNCT ejde-67 74 6 → → PUNCT ejde-67 74 7 x x X ejde-67 74 8 is be AUX ejde-67 74 9 a a DET ejde-67 74 10 continuous continuous ADJ ejde-67 74 11 function function NOUN ejde-67 74 12 and and CCONJ ejde-67 74 13 the the DET ejde-67 74 14 restriction restriction NOUN ejde-67 74 15 of of ADP ejde-67 74 16 x x X ejde-67 74 17 ( ( PUNCT ejde-67 74 18 · · PUNCT ejde-67 74 19 ) ) PUNCT ejde-67 74 20 on on ADP ejde-67 74 21 [ [ X ejde-67 74 22 σ,∞ σ,∞ NOUN ejde-67 74 23 ) ) PUNCT ejde-67 74 24 satisfies satisfy VERB ejde-67 74 25 the the DET ejde-67 74 26 integral integral ADJ ejde-67 74 27 equation equation NOUN ejde-67 74 28 x(t x(t PROPN ejde-67 74 29 ) ) PUNCT ejde-67 75 1 = = SYM ejde-67 75 2 t t PROPN ejde-67 75 3 ( ( PUNCT ejde-67 75 4 t− t− PROPN ejde-67 75 5 σ)x(σ σ)x(σ PROPN ejde-67 75 6 ) ) PUNCT ejde-67 76 1 + + CCONJ ejde-67 76 2 ∫ ∫ PROPN ejde-67 76 3 t t PROPN ejde-67 76 4 σ σ PROPN ejde-67 76 5 t t PROPN ejde-67 76 6 ( ( PUNCT ejde-67 76 7 t− t− PROPN ejde-67 76 8 s)(l(xs s)(l(xs PROPN ejde-67 76 9 ) ) PUNCT ejde-67 76 10 + + NUM ejde-67 76 11 f(s f( NOUN ejde-67 76 12 ) ) PUNCT ejde-67 76 13 ) ) PUNCT ejde-67 77 1 ds ds PROPN ejde-67 77 2 t t PROPN ejde-67 77 3 ≥ ≥ PROPN ejde-67 77 4 σ σ X ejde-67 77 5 . . PUNCT ejde-67 77 6 to to PART ejde-67 77 7 develop develop VERB ejde-67 77 8 new new ADJ ejde-67 77 9 fundamental fundamental ADJ ejde-67 77 10 results result NOUN ejde-67 77 11 about about ADP ejde-67 77 12 the the DET ejde-67 77 13 spectral spectral ADJ ejde-67 77 14 analysis analysis NOUN ejde-67 77 15 of of ADP ejde-67 77 16 the the DET ejde-67 77 17 solutions solution NOUN ejde-67 77 18 , , PUNCT ejde-67 77 19 we we PRON ejde-67 77 20 need need VERB ejde-67 77 21 to to PART ejde-67 77 22 introduce introduce VERB ejde-67 77 23 some some DET ejde-67 77 24 preliminary preliminary ADJ ejde-67 77 25 results result NOUN ejde-67 77 26 . . PUNCT ejde-67 78 1 definition definition NOUN ejde-67 78 2 2.1 2.1 NUM ejde-67 78 3 ( ( PUNCT ejde-67 78 4 [ [ X ejde-67 78 5 10 10 NUM ejde-67 78 6 ] ] NUM ejde-67 78 7 ) ) PUNCT ejde-67 78 8 . . PUNCT ejde-67 79 1 a a DET ejde-67 79 2 c0 c0 NOUN ejde-67 79 3 - - PUNCT ejde-67 79 4 semigroup semigroup PROPN ejde-67 79 5 ( ( PUNCT ejde-67 79 6 t t PROPN ejde-67 79 7 ( ( PUNCT ejde-67 79 8 t))t≥0 t))t≥0 VERB ejde-67 79 9 on on ADP ejde-67 79 10 a a DET ejde-67 79 11 banach banach NOUN ejde-67 79 12 space space NOUN ejde-67 79 13 x x VERB ejde-67 79 14 is be AUX ejde-67 79 15 called call VERB ejde-67 79 16 quasi quasi ADJ ejde-67 79 17 - - ADJ ejde-67 79 18 compact compact ADJ ejde-67 79 19 if if SCONJ ejde-67 79 20 lim lim PROPN ejde-67 79 21 t→+∞ t→+∞ PROPN ejde-67 79 22 d(t d(t PROPN ejde-67 79 23 ( ( PUNCT ejde-67 79 24 t),k(x t),k(x PROPN ejde-67 79 25 ) ) PUNCT ejde-67 79 26 ) ) PUNCT ejde-67 80 1 = = PUNCT ejde-67 80 2 0 0 X ejde-67 80 3 . . PUNCT ejde-67 81 1 definition definition NOUN ejde-67 81 2 2.2 2.2 NUM ejde-67 81 3 . . PUNCT ejde-67 82 1 [ [ X ejde-67 82 2 19 19 NUM ejde-67 82 3 ] ] X ejde-67 82 4 ] ] X ejde-67 82 5 if if SCONJ ejde-67 82 6 b b PROPN ejde-67 82 7 is be AUX ejde-67 82 8 a a DET ejde-67 82 9 bounded bound VERB ejde-67 82 10 set set NOUN ejde-67 82 11 in in ADP ejde-67 82 12 a a DET ejde-67 82 13 banach banach NOUN ejde-67 82 14 space space NOUN ejde-67 82 15 x x NOUN ejde-67 82 16 , , PUNCT ejde-67 82 17 the the DET ejde-67 82 18 kuratowski kuratowski ADJ ejde-67 82 19 measure measure NOUN ejde-67 82 20 of of ADP ejde-67 82 21 noncompactness noncompactness ADJ ejde-67 82 22 is be AUX ejde-67 82 23 defined define VERB ejde-67 82 24 by by ADP ejde-67 82 25 α(b α(b NOUN ejde-67 82 26 ) ) PUNCT ejde-67 83 1 = = SYM ejde-67 83 2 inf{d inf{d ADJ ejde-67 83 3 : : PUNCT ejde-67 83 4 b b X ejde-67 83 5 has have VERB ejde-67 83 6 a a DET ejde-67 83 7 finite finite ADJ ejde-67 83 8 cover cover NOUN ejde-67 83 9 of of ADP ejde-67 83 10 radius radius NOUN ejde-67 83 11 less less ADJ ejde-67 83 12 than than ADP ejde-67 83 13 d d PROPN ejde-67 83 14 } } PUNCT ejde-67 83 15 . . PUNCT ejde-67 84 1 theorem theorem VERB ejde-67 84 2 2.3 2.3 NUM ejde-67 84 3 ( ( PUNCT ejde-67 84 4 [ [ X ejde-67 84 5 19 19 NUM ejde-67 84 6 ] ] NUM ejde-67 84 7 ) ) PUNCT ejde-67 84 8 . . PUNCT ejde-67 85 1 assume assume VERB ejde-67 85 2 that that SCONJ ejde-67 85 3 x x PRON ejde-67 85 4 is be AUX ejde-67 85 5 a a DET ejde-67 85 6 banach banach NOUN ejde-67 85 7 space space NOUN ejde-67 85 8 and and CCONJ ejde-67 85 9 α α PRON ejde-67 85 10 ( ( PUNCT ejde-67 85 11 · · PUNCT ejde-67 85 12 ) ) PUNCT ejde-67 85 13 is be AUX ejde-67 85 14 the the DET ejde-67 85 15 the the DET ejde-67 85 16 kuratowski kuratowski ADJ ejde-67 85 17 measure measure NOUN ejde-67 85 18 of of ADP ejde-67 85 19 noncompactness noncompactness NOUN ejde-67 85 20 of of ADP ejde-67 85 21 a a DET ejde-67 85 22 bounded bound VERB ejde-67 85 23 set set NOUN ejde-67 85 24 b b PROPN ejde-67 85 25 of of ADP ejde-67 85 26 x. x. NOUN ejde-67 85 27 then then ADV ejde-67 85 28 ( ( PUNCT ejde-67 85 29 i i NOUN ejde-67 85 30 ) ) PUNCT ejde-67 85 31 α(b α(b NOUN ejde-67 85 32 ) ) PUNCT ejde-67 86 1 = = SYM ejde-67 86 2 0 0 PUNCT ejde-67 87 1 if if SCONJ ejde-67 87 2 and and CCONJ ejde-67 87 3 only only ADV ejde-67 87 4 if if SCONJ ejde-67 87 5 the the DET ejde-67 87 6 closure closure NOUN ejde-67 87 7 of of ADP ejde-67 87 8 b b NOUN ejde-67 87 9 is be AUX ejde-67 87 10 compact compact ADJ ejde-67 87 11 . . PUNCT ejde-67 88 1 4 4 NUM ejde-67 88 2 m. m. NOUN ejde-67 88 3 el el PROPN ejde-67 88 4 attaouy attaouy PROPN ejde-67 88 5 , , PUNCT ejde-67 88 6 k. k. PROPN ejde-67 88 7 ezzinbi ezzinbi PROPN ejde-67 88 8 , , PUNCT ejde-67 88 9 g. g. PROPN ejde-67 88 10 m. m. PROPN ejde-67 88 11 n’guérékata n’guérékata PROPN ejde-67 88 12 ejde-2023/39 ejde-2023/39 PROPN ejde-67 88 13 ( ( PUNCT ejde-67 88 14 ii ii NOUN ejde-67 88 15 ) ) PUNCT ejde-67 88 16 α(a α(a NOUN ejde-67 88 17 ∪ ∪ ADJ ejde-67 88 18 b b NOUN ejde-67 88 19 ) ) PUNCT ejde-67 88 20 = = SYM ejde-67 88 21 max(α(a max(α(a NOUN ejde-67 88 22 ) ) PUNCT ejde-67 88 23 , , PUNCT ejde-67 88 24 α(b α(b NOUN ejde-67 88 25 ) ) PUNCT ejde-67 88 26 ) ) PUNCT ejde-67 88 27 . . PUNCT ejde-67 89 1 ( ( PUNCT ejde-67 89 2 iii iii NOUN ejde-67 89 3 ) ) PUNCT ejde-67 89 4 α(a+b α(a+b NOUN ejde-67 89 5 ) ) PUNCT ejde-67 89 6 ≤ ≤ NOUN ejde-67 89 7 α(a α(a NOUN ejde-67 89 8 ) ) PUNCT ejde-67 89 9 + + CCONJ ejde-67 89 10 α(b α(b NOUN ejde-67 89 11 ) ) PUNCT ejde-67 89 12 . . PUNCT ejde-67 90 1 ( ( PUNCT ejde-67 90 2 iv iv X ejde-67 90 3 ) ) PUNCT ejde-67 90 4 α(cob α(cob PROPN ejde-67 90 5 ) ) PUNCT ejde-67 90 6 = = SYM ejde-67 90 7 α(b α(b NOUN ejde-67 90 8 ) ) PUNCT ejde-67 90 9 where where SCONJ ejde-67 90 10 cob cob NOUN ejde-67 90 11 is be AUX ejde-67 90 12 the the DET ejde-67 90 13 closed closed ADJ ejde-67 90 14 convex convex NOUN ejde-67 90 15 hull hull NOUN ejde-67 90 16 of of ADP ejde-67 90 17 b. b. PROPN ejde-67 90 18 definition definition NOUN ejde-67 90 19 2.4 2.4 NUM ejde-67 90 20 ( ( PUNCT ejde-67 90 21 [ [ X ejde-67 90 22 10 10 NUM ejde-67 90 23 ] ] NUM ejde-67 90 24 ) ) PUNCT ejde-67 90 25 . . PUNCT ejde-67 91 1 let let VERB ejde-67 91 2 s s PRON ejde-67 91 3 ∈ ∈ VERB ejde-67 91 4 l(x l(x PROPN ejde-67 91 5 ) ) PUNCT ejde-67 91 6 . . PUNCT ejde-67 92 1 the the DET ejde-67 92 2 essential essential ADJ ejde-67 92 3 norm norm NOUN ejde-67 92 4 is be AUX ejde-67 92 5 defined define VERB ejde-67 92 6 by by ADP ejde-67 92 7 |s|ess |s|ess ADP ejde-67 92 8 = = PUNCT ejde-67 92 9 inf{c inf{c X ejde-67 92 10 > > X ejde-67 92 11 0 0 NUM ejde-67 92 12 : : PUNCT ejde-67 92 13 α(s(b α(s(b NUM ejde-67 92 14 ) ) PUNCT ejde-67 92 15 ) ) PUNCT ejde-67 93 1 ≤ ≤ NOUN ejde-67 93 2 cα(b cα(b PUNCT ejde-67 93 3 ) ) PUNCT ejde-67 93 4 for for ADP ejde-67 93 5 any any DET ejde-67 93 6 bounded bounded PROPN ejde-67 93 7 b b PROPN ejde-67 93 8 ∈ ∈ PROPN ejde-67 93 9 x x X ejde-67 93 10 } } PUNCT ejde-67 93 11 . . PUNCT ejde-67 94 1 definition definition NOUN ejde-67 94 2 2.5 2.5 NUM ejde-67 94 3 ( ( PUNCT ejde-67 94 4 [ [ X ejde-67 94 5 10 10 NUM ejde-67 94 6 ] ] NUM ejde-67 94 7 ) ) PUNCT ejde-67 94 8 . . PUNCT ejde-67 95 1 the the DET ejde-67 95 2 essential essential ADJ ejde-67 95 3 growth growth NOUN ejde-67 95 4 bound bind VERB ejde-67 95 5 of of ADP ejde-67 95 6 a a DET ejde-67 95 7 c0 c0 NOUN ejde-67 95 8 - - PUNCT ejde-67 95 9 semigroup semigroup PROPN ejde-67 95 10 ( ( PUNCT ejde-67 95 11 t t PROPN ejde-67 95 12 ( ( PUNCT ejde-67 95 13 t))t≥0 t))t≥0 PROPN ejde-67 95 14 is be AUX ejde-67 95 15 defined define VERB ejde-67 95 16 by by ADP ejde-67 95 17 wess wess NOUN ejde-67 95 18 = = PUNCT ejde-67 95 19 inf{w inf{w PROPN ejde-67 95 20 ∈ ∈ NOUN ejde-67 95 21 r r NOUN ejde-67 95 22 : : PUNCT ejde-67 95 23 sup sup PROPN ejde-67 95 24 t≥0 t≥0 NOUN ejde-67 95 25 e−wt|t e−wt|t NUM ejde-67 95 26 ( ( PUNCT ejde-67 95 27 t)|ess t)|ess PROPN ejde-67 95 28 < < X ejde-67 95 29 ∞ ∞ NUM ejde-67 95 30 } } PUNCT ejde-67 95 31 . . PUNCT ejde-67 96 1 theorem theorem VERB ejde-67 96 2 2.6 2.6 NUM ejde-67 96 3 ( ( PUNCT ejde-67 96 4 [ [ X ejde-67 96 5 10 10 NUM ejde-67 96 6 ] ] NUM ejde-67 96 7 ) ) PUNCT ejde-67 96 8 . . PUNCT ejde-67 97 1 for for ADP ejde-67 97 2 a a DET ejde-67 97 3 c0 c0 NOUN ejde-67 97 4 - - PUNCT ejde-67 97 5 semigroup semigroup PROPN ejde-67 97 6 ( ( PUNCT ejde-67 97 7 t t PROPN ejde-67 97 8 ( ( PUNCT ejde-67 97 9 t))t≥0 t))t≥0 VERB ejde-67 97 10 on on ADP ejde-67 97 11 a a DET ejde-67 97 12 banach banach NOUN ejde-67 97 13 space space NOUN ejde-67 97 14 x x NOUN ejde-67 97 15 , , PUNCT ejde-67 97 16 the the DET ejde-67 97 17 following follow VERB ejde-67 97 18 assertions assertion NOUN ejde-67 97 19 are be AUX ejde-67 97 20 equivalent equivalent ADJ ejde-67 97 21 : : PUNCT ejde-67 97 22 ( ( PUNCT ejde-67 97 23 i i NOUN ejde-67 97 24 ) ) PUNCT ejde-67 97 25 ( ( PUNCT ejde-67 97 26 t t PROPN ejde-67 97 27 ( ( PUNCT ejde-67 97 28 t))t≥0 t))t≥0 PROPN ejde-67 97 29 is be AUX ejde-67 97 30 quasi quasi ADJ ejde-67 97 31 - - ADJ ejde-67 97 32 compact compact ADJ ejde-67 97 33 . . PUNCT ejde-67 98 1 ( ( PUNCT ejde-67 98 2 ii ii X ejde-67 98 3 ) ) PUNCT ejde-67 98 4 wess wess NOUN ejde-67 98 5 < < X ejde-67 98 6 0 0 NUM ejde-67 98 7 . . PUNCT ejde-67 99 1 ( ( PUNCT ejde-67 99 2 iii iii NOUN ejde-67 99 3 ) ) PUNCT ejde-67 99 4 ‖t ‖t NOUN ejde-67 99 5 ( ( PUNCT ejde-67 99 6 t0)−k‖ t0)−k‖ VERB ejde-67 99 7 < < X ejde-67 99 8 1 1 NUM ejde-67 99 9 , , PUNCT ejde-67 99 10 for for ADP ejde-67 99 11 some some DET ejde-67 99 12 t0 t0 PROPN ejde-67 99 13 > > X ejde-67 99 14 0 0 PUNCT ejde-67 100 1 and and CCONJ ejde-67 100 2 k k PROPN ejde-67 100 3 ∈ ∈ PROPN ejde-67 100 4 k(x k(x PROPN ejde-67 100 5 ) ) PUNCT ejde-67 100 6 . . PUNCT ejde-67 101 1 now now ADV ejde-67 101 2 , , PUNCT ejde-67 101 3 we we PRON ejde-67 101 4 consider consider VERB ejde-67 101 5 the the DET ejde-67 101 6 linear linear ADJ ejde-67 101 7 problem problem NOUN ejde-67 101 8 x′(t x′(t NOUN ejde-67 101 9 ) ) PUNCT ejde-67 101 10 = = SYM ejde-67 101 11 ax(t ax(t NUM ejde-67 101 12 ) ) PUNCT ejde-67 102 1 + + CCONJ ejde-67 102 2 l(xt l(xt NOUN ejde-67 102 3 ) ) PUNCT ejde-67 102 4 , , PUNCT ejde-67 102 5 for for ADP ejde-67 102 6 t t PROPN ejde-67 102 7 ≥ ≥ NOUN ejde-67 102 8 0 0 NUM ejde-67 102 9 , , PUNCT ejde-67 102 10 x0 x0 PROPN ejde-67 102 11 = = PUNCT ejde-67 102 12 ϕ ϕ PROPN ejde-67 102 13 ∈ ∈ PROPN ejde-67 102 14 c. c. NOUN ejde-67 102 15 ( ( PUNCT ejde-67 102 16 2.2 2.2 NUM ejde-67 102 17 ) ) PUNCT ejde-67 102 18 the the DET ejde-67 102 19 solution solution NOUN ejde-67 102 20 operator operator NOUN ejde-67 102 21 v v NOUN ejde-67 102 22 ( ( PUNCT ejde-67 102 23 t t PROPN ejde-67 102 24 ) ) PUNCT ejde-67 102 25 is be AUX ejde-67 102 26 defined define VERB ejde-67 102 27 by by ADP ejde-67 102 28 v v NOUN ejde-67 102 29 ( ( PUNCT ejde-67 102 30 t)ϕ t)ϕ ADJ ejde-67 102 31 = = SYM ejde-67 102 32 xt xt X ejde-67 102 33 ( ( PUNCT ejde-67 102 34 · · PUNCT ejde-67 102 35 , , PUNCT ejde-67 102 36 ϕ ϕ NOUN ejde-67 102 37 ) ) PUNCT ejde-67 102 38 , , PUNCT ejde-67 102 39 where where SCONJ ejde-67 102 40 x x X ejde-67 102 41 ( ( PUNCT ejde-67 102 42 · · PUNCT ejde-67 102 43 , , PUNCT ejde-67 102 44 ϕ ϕ NOUN ejde-67 102 45 ) ) PUNCT ejde-67 102 46 is be AUX ejde-67 102 47 the the DET ejde-67 102 48 mild mild ADJ ejde-67 102 49 solution solution NOUN ejde-67 102 50 of of ADP ejde-67 102 51 ( ( PUNCT ejde-67 102 52 2.2 2.2 NUM ejde-67 102 53 ) ) PUNCT ejde-67 102 54 . . PUNCT ejde-67 103 1 for for SCONJ ejde-67 103 2 more more ADJ ejde-67 103 3 details detail NOUN ejde-67 103 4 see see VERB ejde-67 103 5 [ [ X ejde-67 103 6 19 19 NUM ejde-67 103 7 ] ] PUNCT ejde-67 103 8 . . PUNCT ejde-67 104 1 theorem theorem ADJ ejde-67 104 2 2.7 2.7 NUM ejde-67 104 3 ( ( PUNCT ejde-67 104 4 [ [ X ejde-67 104 5 12 12 NUM ejde-67 104 6 ] ] PUNCT ejde-67 104 7 ) ) PUNCT ejde-67 104 8 . . PUNCT ejde-67 105 1 ( ( PUNCT ejde-67 105 2 v v X ejde-67 105 3 ( ( PUNCT ejde-67 105 4 t))t≥0 t))t≥0 PROPN ejde-67 105 5 is be AUX ejde-67 105 6 a a DET ejde-67 105 7 c0 c0 NOUN ejde-67 105 8 - - PUNCT ejde-67 105 9 semigroup semigroup NOUN ejde-67 105 10 of of ADP ejde-67 105 11 bounded bounded ADJ ejde-67 105 12 linear linear PROPN ejde-67 105 13 operators operator NOUN ejde-67 105 14 on on ADP ejde-67 105 15 c c PROPN ejde-67 105 16 , , PUNCT ejde-67 105 17 the the DET ejde-67 105 18 infinitesimal infinitesimal ADJ ejde-67 105 19 generator generator NOUN ejde-67 105 20 a a NOUN ejde-67 105 21 is be AUX ejde-67 105 22 given give VERB ejde-67 105 23 by by ADP ejde-67 105 24 d(a d(a PROPN ejde-67 105 25 ) ) PUNCT ejde-67 106 1 = = PRON ejde-67 106 2 { { PUNCT ejde-67 106 3 ϕ ϕ NOUN ejde-67 106 4 ∈ ∈ PROPN ejde-67 106 5 c1([−r c1([−r PROPN ejde-67 106 6 , , PUNCT ejde-67 106 7 0 0 NUM ejde-67 106 8 ] ] PUNCT ejde-67 106 9 , , PUNCT ejde-67 106 10 x x X ejde-67 106 11 ) ) PUNCT ejde-67 106 12 : : PUNCT ejde-67 107 1 ϕ(0 ϕ(0 PRON ejde-67 107 2 ) ) PUNCT ejde-67 107 3 ∈ ∈ PROPN ejde-67 107 4 d(a)and d(a)and NOUN ejde-67 107 5 ϕ′(0 ϕ′(0 PROPN ejde-67 107 6 ) ) PUNCT ejde-67 108 1 = = SYM ejde-67 108 2 aϕ(0 aϕ(0 PROPN ejde-67 108 3 ) ) PUNCT ejde-67 109 1 + + CCONJ ejde-67 110 1 l(ϕ l(ϕ NOUN ejde-67 110 2 ) ) PUNCT ejde-67 110 3 } } PUNCT ejde-67 110 4 aϕ aϕ NOUN ejde-67 110 5 = = NOUN ejde-67 110 6 ϕ′. ϕ′. PRON ejde-67 110 7 lemma lemma PROPN ejde-67 110 8 2.8 2.8 NUM ejde-67 110 9 ( ( PUNCT ejde-67 110 10 [ [ X ejde-67 110 11 12 12 NUM ejde-67 110 12 ] ] PUNCT ejde-67 110 13 ) ) PUNCT ejde-67 110 14 . . PUNCT ejde-67 111 1 assume assume VERB ejde-67 111 2 that that SCONJ ejde-67 111 3 ( ( PUNCT ejde-67 111 4 h1 h1 NOUN ejde-67 111 5 ) ) PUNCT ejde-67 111 6 holds hold VERB ejde-67 111 7 . . PUNCT ejde-67 112 1 then then ADV ejde-67 112 2 [ [ X ejde-67 112 3 v v X ejde-67 112 4 ( ( PUNCT ejde-67 112 5 t)ϕ](θ t)ϕ](θ NOUN ejde-67 112 6 ) ) PUNCT ejde-67 112 7 = = PRON ejde-67 112 8 { { PUNCT ejde-67 113 1 [ [ X ejde-67 113 2 v v X ejde-67 113 3 ( ( PUNCT ejde-67 113 4 t+ t+ NOUN ejde-67 113 5 θ)ϕ](0 θ)ϕ](0 NOUN ejde-67 113 6 ) ) PUNCT ejde-67 113 7 , , PUNCT ejde-67 113 8 t+ t+ PUNCT ejde-67 113 9 θ θ PROPN ejde-67 113 10 ≥ ≥ NUM ejde-67 113 11 0 0 NUM ejde-67 113 12 , , PUNCT ejde-67 113 13 ϕ(t+ ϕ(t+ NOUN ejde-67 113 14 θ θ NOUN ejde-67 113 15 ) ) PUNCT ejde-67 113 16 , , PUNCT ejde-67 113 17 t+ t+ VERB ejde-67 113 18 θ θ PROPN ejde-67 113 19 ≤ ≤ ADV ejde-67 113 20 0 0 NUM ejde-67 113 21 . . PUNCT ejde-67 114 1 let let VERB ejde-67 114 2 w w PROPN ejde-67 114 3 ( ( PUNCT ejde-67 114 4 t t PROPN ejde-67 114 5 ) ) PUNCT ejde-67 114 6 the the DET ejde-67 114 7 solution solution NOUN ejde-67 114 8 operator operator NOUN ejde-67 114 9 corresponding correspond VERB ejde-67 114 10 to to ADP ejde-67 114 11 l l NOUN ejde-67 114 12 = = PUNCT ejde-67 115 1 0 0 X ejde-67 115 2 . . PUNCT ejde-67 116 1 then then ADV ejde-67 116 2 w w PROPN ejde-67 116 3 ( ( PUNCT ejde-67 116 4 t t PROPN ejde-67 116 5 ) ) PUNCT ejde-67 116 6 is be AUX ejde-67 116 7 given give VERB ejde-67 116 8 by by ADP ejde-67 116 9 [ [ X ejde-67 116 10 w w X ejde-67 116 11 ( ( PUNCT ejde-67 116 12 t)ϕ](θ t)ϕ](θ INTJ ejde-67 116 13 ) ) PUNCT ejde-67 116 14 = = PRON ejde-67 116 15 { { PUNCT ejde-67 116 16 t t PROPN ejde-67 116 17 ( ( PUNCT ejde-67 116 18 t+ t+ NOUN ejde-67 116 19 θ)ϕ(0 θ)ϕ(0 NOUN ejde-67 116 20 ) ) PUNCT ejde-67 116 21 , , PUNCT ejde-67 116 22 −t −t VERB ejde-67 116 23 ≤ ≤ NUM ejde-67 116 24 θ θ PROPN ejde-67 116 25 ≤ ≤ NUM ejde-67 116 26 0 0 NUM ejde-67 116 27 , , PUNCT ejde-67 116 28 ϕ(t+ ϕ(t+ NOUN ejde-67 116 29 θ θ NOUN ejde-67 116 30 ) ) PUNCT ejde-67 116 31 , , PUNCT ejde-67 116 32 −r −r ADJ ejde-67 116 33 ≤ ≤ NUM ejde-67 116 34 θ θ PROPN ejde-67 116 35 ≤ ≤ PROPN ejde-67 116 36 −t −t NOUN ejde-67 116 37 . . PUNCT ejde-67 117 1 we we PRON ejde-67 117 2 establish establish VERB ejde-67 117 3 the the DET ejde-67 117 4 first first ADJ ejde-67 117 5 result result NOUN ejde-67 117 6 on on ADP ejde-67 117 7 the the DET ejde-67 117 8 asymptotic asymptotic ADJ ejde-67 117 9 behavior behavior NOUN ejde-67 117 10 of of ADP ejde-67 117 11 the the DET ejde-67 117 12 semigroup semigroup NOUN ejde-67 117 13 ( ( PUNCT ejde-67 117 14 v v NOUN ejde-67 117 15 ( ( PUNCT ejde-67 117 16 t))t≥0 t))t≥0 NOUN ejde-67 117 17 by by ADP ejde-67 117 18 the the DET ejde-67 117 19 following follow VERB ejde-67 117 20 theorem theorem PROPN ejde-67 117 21 . . PUNCT ejde-67 117 22 theorem theorem VERB ejde-67 117 23 2.9 2.9 NUM ejde-67 117 24 ( ( PUNCT ejde-67 117 25 [ [ X ejde-67 117 26 12 12 NUM ejde-67 117 27 ] ] PUNCT ejde-67 117 28 ) ) PUNCT ejde-67 117 29 . . PUNCT ejde-67 118 1 assume assume VERB ejde-67 118 2 that that SCONJ ejde-67 118 3 the the DET ejde-67 118 4 semigroup semigroup NOUN ejde-67 118 5 ( ( PUNCT ejde-67 118 6 t t PROPN ejde-67 118 7 ( ( PUNCT ejde-67 118 8 t))t≥0 t))t≥0 PROPN ejde-67 118 9 is be AUX ejde-67 118 10 exponentially exponentially ADV ejde-67 118 11 stable stable ADJ ejde-67 118 12 and and CCONJ ejde-67 118 13 that that SCONJ ejde-67 118 14 the the DET ejde-67 118 15 operator operator NOUN ejde-67 118 16 t t NOUN ejde-67 118 17 ( ( PUNCT ejde-67 118 18 t)l t)l NOUN ejde-67 118 19 : : PUNCT ejde-67 118 20 c c X ejde-67 118 21 → → PUNCT ejde-67 118 22 x x X ejde-67 118 23 is be AUX ejde-67 118 24 compact compact ADJ ejde-67 118 25 for for ADP ejde-67 118 26 all all DET ejde-67 118 27 t t PROPN ejde-67 118 28 > > X ejde-67 118 29 0 0 X ejde-67 118 30 . . PUNCT ejde-67 119 1 then then ADV ejde-67 119 2 , , PUNCT ejde-67 119 3 the the DET ejde-67 119 4 semigroup semigroup NOUN ejde-67 119 5 ( ( PUNCT ejde-67 119 6 v v NOUN ejde-67 119 7 ( ( PUNCT ejde-67 119 8 t))t≥0 t))t≥0 PROPN ejde-67 119 9 is be AUX ejde-67 119 10 quasi quasi ADJ ejde-67 119 11 - - ADJ ejde-67 119 12 compact compact ADJ ejde-67 119 13 . . PUNCT ejde-67 120 1 remark remark PROPN ejde-67 120 2 2.10 2.10 NUM ejde-67 120 3 . . PUNCT ejde-67 121 1 ( ( PUNCT ejde-67 121 2 i i NOUN ejde-67 121 3 ) ) PUNCT ejde-67 121 4 the the DET ejde-67 121 5 operator operator NOUN ejde-67 121 6 t t PROPN ejde-67 121 7 ( ( PUNCT ejde-67 121 8 t)l t)l NOUN ejde-67 121 9 is be AUX ejde-67 121 10 compact compact ADJ ejde-67 121 11 if if SCONJ ejde-67 121 12 the the DET ejde-67 121 13 semigroup semigroup NOUN ejde-67 121 14 ( ( PUNCT ejde-67 121 15 t t PROPN ejde-67 121 16 ( ( PUNCT ejde-67 121 17 t))t≥0 t))t≥0 PROPN ejde-67 121 18 is be AUX ejde-67 121 19 compact compact ADJ ejde-67 121 20 or or CCONJ ejde-67 121 21 the the DET ejde-67 121 22 linear linear ADJ ejde-67 121 23 delay delay NOUN ejde-67 121 24 operator operator NOUN ejde-67 121 25 l l NOUN ejde-67 121 26 is be AUX ejde-67 121 27 compact compact ADJ ejde-67 121 28 . . PUNCT ejde-67 122 1 if if SCONJ ejde-67 122 2 for for ADP ejde-67 122 3 example example NOUN ejde-67 122 4 ( ( PUNCT ejde-67 122 5 t t PROPN ejde-67 122 6 ( ( PUNCT ejde-67 122 7 t))t≥0 t))t≥0 PROPN ejde-67 122 8 is be AUX ejde-67 122 9 not not PART ejde-67 122 10 necessarily necessarily ADV ejde-67 122 11 compact compact ADJ ejde-67 122 12 and and CCONJ ejde-67 122 13 l l NOUN ejde-67 122 14 is be AUX ejde-67 122 15 given give VERB ejde-67 122 16 by by ADP ejde-67 122 17 l(ϕ l(ϕ NUM ejde-67 122 18 ) ) PUNCT ejde-67 122 19 = = SYM ejde-67 123 1 ∑k ∑k PROPN ejde-67 123 2 i=1biϕ(−ri i=1biϕ(−ri PROPN ejde-67 123 3 ) ) PUNCT ejde-67 123 4 , , PUNCT ejde-67 123 5 where where SCONJ ejde-67 123 6 bi bi NOUN ejde-67 123 7 : : PUNCT ejde-67 123 8 x x SYM ejde-67 123 9 → → SYM ejde-67 123 10 x x X ejde-67 123 11 , , PUNCT ejde-67 123 12 for for ADP ejde-67 123 13 i i PROPN ejde-67 123 14 = = NOUN ejde-67 123 15 1 1 NUM ejde-67 123 16 , , PUNCT ejde-67 123 17 . . PUNCT ejde-67 123 18 . . PUNCT ejde-67 123 19 . . PUNCT ejde-67 124 1 , , PUNCT ejde-67 124 2 k k PROPN ejde-67 124 3 are be AUX ejde-67 124 4 compact compact ADJ ejde-67 124 5 linear linear ADJ ejde-67 124 6 operators operator NOUN ejde-67 124 7 on on ADP ejde-67 124 8 x x NOUN ejde-67 124 9 , , PUNCT ejde-67 124 10 then then ADV ejde-67 124 11 of of ADP ejde-67 124 12 course course NOUN ejde-67 124 13 t t NOUN ejde-67 124 14 ( ( PUNCT ejde-67 124 15 t)l t)l NOUN ejde-67 124 16 is be AUX ejde-67 124 17 compact compact ADJ ejde-67 124 18 . . PUNCT ejde-67 125 1 ejde-2023/39 ejde-2023/39 ADJ ejde-67 125 2 reduction reduction NOUN ejde-67 125 3 principle principle NOUN ejde-67 125 4 5 5 NUM ejde-67 125 5 ( ( PUNCT ejde-67 125 6 ii ii NOUN ejde-67 125 7 ) ) PUNCT ejde-67 125 8 if if SCONJ ejde-67 125 9 the the DET ejde-67 125 10 semigroup semigroup NOUN ejde-67 125 11 ( ( PUNCT ejde-67 125 12 t t PROPN ejde-67 125 13 ( ( PUNCT ejde-67 125 14 t))t≥0 t))t≥0 PROPN ejde-67 125 15 is be AUX ejde-67 125 16 not not PART ejde-67 125 17 exponentially exponentially ADV ejde-67 125 18 stable stable ADJ ejde-67 125 19 , , PUNCT ejde-67 125 20 we we PRON ejde-67 125 21 can can AUX ejde-67 125 22 substitute substitute VERB ejde-67 125 23 the the DET ejde-67 125 24 operator operator NOUN ejde-67 125 25 a a PRON ejde-67 125 26 by by ADP ejde-67 125 27 a a DET ejde-67 125 28 − − NOUN ejde-67 125 29 αi αi NOUN ejde-67 125 30 , , PUNCT ejde-67 125 31 where where SCONJ ejde-67 125 32 α α PROPN ejde-67 125 33 is be AUX ejde-67 125 34 an an DET ejde-67 125 35 enough enough ADJ ejde-67 125 36 large large ADJ ejde-67 125 37 constant constant ADJ ejde-67 125 38 . . PUNCT ejde-67 126 1 then then ADV ejde-67 126 2 , , PUNCT ejde-67 126 3 we we PRON ejde-67 126 4 obtain obtain VERB ejde-67 126 5 that that SCONJ ejde-67 126 6 the the DET ejde-67 126 7 semigroup semigroup PROPN ejde-67 126 8 e−αt(t e−αt(t NOUN ejde-67 126 9 ( ( PUNCT ejde-67 126 10 t))t≥0 t))t≥0 PROPN ejde-67 126 11 is be AUX ejde-67 126 12 exponentially exponentially ADV ejde-67 126 13 stable stable ADJ ejde-67 126 14 and and CCONJ ejde-67 126 15 we we PRON ejde-67 126 16 assume assume VERB ejde-67 126 17 that that SCONJ ejde-67 126 18 the the DET ejde-67 126 19 operator operator NOUN ejde-67 126 20 l+ l+ PUNCT ejde-67 126 21 αi αi PRON ejde-67 126 22 is be AUX ejde-67 126 23 compact compact ADJ ejde-67 126 24 . . PUNCT ejde-67 127 1 to to PART ejde-67 127 2 give give VERB ejde-67 127 3 the the DET ejde-67 127 4 variation variation NOUN ejde-67 127 5 of of ADP ejde-67 127 6 constants constant NOUN ejde-67 127 7 formula formula NOUN ejde-67 127 8 , , PUNCT ejde-67 127 9 we we PRON ejde-67 127 10 need need VERB ejde-67 127 11 to to PART ejde-67 127 12 recall recall VERB ejde-67 127 13 some some DET ejde-67 127 14 notation notation NOUN ejde-67 127 15 and and CCONJ ejde-67 127 16 results result NOUN ejde-67 127 17 which which PRON ejde-67 127 18 are be AUX ejde-67 127 19 taken take VERB ejde-67 127 20 from from ADP ejde-67 127 21 [ [ X ejde-67 127 22 11 11 NUM ejde-67 127 23 ] ] PUNCT ejde-67 127 24 . . PUNCT ejde-67 128 1 let let VERB ejde-67 128 2 〈 〈 PROPN ejde-67 128 3 x0 x0 PROPN ejde-67 128 4 〉 〉 PROPN ejde-67 128 5 be be AUX ejde-67 128 6 the the DET ejde-67 128 7 space space NOUN ejde-67 128 8 defined define VERB ejde-67 128 9 by by ADP ejde-67 128 10 〈 〈 PROPN ejde-67 128 11 x0 x0 PROPN ejde-67 128 12 〉 〉 NOUN ejde-67 128 13 = = SYM ejde-67 129 1 { { PUNCT ejde-67 129 2 x0c x0c NOUN ejde-67 129 3 : : PUNCT ejde-67 129 4 c c X ejde-67 129 5 ∈ ∈ PROPN ejde-67 129 6 x x X ejde-67 129 7 } } PUNCT ejde-67 129 8 , , PUNCT ejde-67 129 9 where where SCONJ ejde-67 129 10 ( ( PUNCT ejde-67 129 11 x0c)(θ x0c)(θ PROPN ejde-67 129 12 ) ) PUNCT ejde-67 130 1 = = PRON ejde-67 130 2 { { PUNCT ejde-67 130 3 0 0 NUM ejde-67 130 4 , , PUNCT ejde-67 130 5 if if SCONJ ejde-67 130 6 θ θ PROPN ejde-67 130 7 ∈ ∈ PROPN ejde-67 131 1 [ [ X ejde-67 131 2 −r −r ADJ ejde-67 131 3 , , PUNCT ejde-67 131 4 0 0 NUM ejde-67 131 5 ] ] X ejde-67 131 6 c c X ejde-67 131 7 , , PUNCT ejde-67 131 8 if if SCONJ ejde-67 131 9 θ θ PROPN ejde-67 131 10 = = SYM ejde-67 131 11 0 0 X ejde-67 131 12 . . PUNCT ejde-67 132 1 the the DET ejde-67 132 2 space space NOUN ejde-67 132 3 c c PROPN ejde-67 132 4 ⊕ ⊕ PROPN ejde-67 132 5 〈 〈 PROPN ejde-67 132 6 x0 x0 PROPN ejde-67 132 7 〉 〉 PROPN ejde-67 132 8 is be AUX ejde-67 132 9 equipped equip VERB ejde-67 132 10 with with ADP ejde-67 132 11 the the DET ejde-67 132 12 norm norm NOUN ejde-67 132 13 |φ+x0c| |φ+x0c| NOUN ejde-67 132 14 = = SYM ejde-67 132 15 |φ|c |φ|c PROPN ejde-67 132 16 + + CCONJ ejde-67 132 17 |c| |c| PROPN ejde-67 132 18 for for ADP ejde-67 132 19 ( ( PUNCT ejde-67 132 20 φ φ PROPN ejde-67 132 21 , , PUNCT ejde-67 132 22 c c NOUN ejde-67 132 23 ) ) PUNCT ejde-67 132 24 ∈ ∈ PROPN ejde-67 132 25 c c PROPN ejde-67 132 26 ×x ×x X ejde-67 132 27 , , PUNCT ejde-67 132 28 is be AUX ejde-67 132 29 a a DET ejde-67 132 30 banach banach NOUN ejde-67 132 31 space space NOUN ejde-67 132 32 and and CCONJ ejde-67 132 33 consider consider VERB ejde-67 132 34 the the DET ejde-67 132 35 extension extension NOUN ejde-67 132 36 ã ã PROPN ejde-67 132 37 of of ADP ejde-67 132 38 the the DET ejde-67 132 39 operator operator NOUN ejde-67 133 1 a a PRON ejde-67 133 2 defined define VERB ejde-67 133 3 on on ADP ejde-67 133 4 c c PROPN ejde-67 133 5 ⊕ ⊕ PROPN ejde-67 133 6 〈 〈 PROPN ejde-67 133 7 x0 x0 PROPN ejde-67 133 8 〉 〉 PROPN ejde-67 133 9 by by ADP ejde-67 133 10 d(ã d(ã PROPN ejde-67 133 11 ) ) PUNCT ejde-67 133 12 = = PRON ejde-67 134 1 { { PUNCT ejde-67 134 2 ϕ ϕ NOUN ejde-67 134 3 ∈ ∈ PROPN ejde-67 134 4 c1([−r c1([−r PROPN ejde-67 134 5 , , PUNCT ejde-67 134 6 0 0 NUM ejde-67 134 7 ] ] PUNCT ejde-67 134 8 , , PUNCT ejde-67 134 9 x x NOUN ejde-67 134 10 ) ) PUNCT ejde-67 134 11 , , PUNCT ejde-67 134 12 ϕ(0 ϕ(0 PROPN ejde-67 134 13 ) ) PUNCT ejde-67 134 14 ∈ ∈ PROPN ejde-67 135 1 d(a d(a PROPN ejde-67 135 2 ) ) PUNCT ejde-67 135 3 and and CCONJ ejde-67 135 4 ϕ′(0 ϕ′(0 NOUN ejde-67 135 5 ) ) PUNCT ejde-67 135 6 ∈ ∈ PROPN ejde-67 135 7 d(a d(a PROPN ejde-67 135 8 ) ) PUNCT ejde-67 135 9 } } PUNCT ejde-67 135 10 , , PUNCT ejde-67 135 11 ãϕ ãϕ ADP ejde-67 135 12 = = PRON ejde-67 135 13 ϕ′ ϕ′ X ejde-67 136 1 + + ADJ ejde-67 136 2 x0(aϕ(0 x0(aϕ(0 NUM ejde-67 136 3 ) ) PUNCT ejde-67 137 1 + + CCONJ ejde-67 137 2 l(ϕ)− l(ϕ)− ADP ejde-67 137 3 ϕ′(0 ϕ′(0 NOUN ejde-67 137 4 ) ) PUNCT ejde-67 137 5 ) ) PUNCT ejde-67 137 6 . . PUNCT ejde-67 138 1 lemma lemma PROPN ejde-67 138 2 2.11 2.11 NUM ejde-67 138 3 ( ( PUNCT ejde-67 138 4 [ [ X ejde-67 138 5 11 11 NUM ejde-67 138 6 ] ] NUM ejde-67 138 7 ) ) PUNCT ejde-67 138 8 . . PUNCT ejde-67 138 9 assume assume VERB ejde-67 138 10 that that SCONJ ejde-67 138 11 ( ( PUNCT ejde-67 138 12 h1 h1 NOUN ejde-67 138 13 ) ) PUNCT ejde-67 138 14 holds hold VERB ejde-67 138 15 . . PUNCT ejde-67 139 1 then then ADV ejde-67 139 2 , , PUNCT ejde-67 139 3 ã ã PROPN ejde-67 139 4 satisfies satisfy VERB ejde-67 139 5 the the DET ejde-67 139 6 hille hille PROPN ejde-67 139 7 - - PUNCT ejde-67 139 8 yosida yosida PROPN ejde-67 139 9 condition condition NOUN ejde-67 139 10 on on ADP ejde-67 139 11 c c PROPN ejde-67 139 12 ⊕ ⊕ PROPN ejde-67 139 13 〈 〈 PROPN ejde-67 139 14 x0 x0 PROPN ejde-67 139 15 〉 〉 PROPN ejde-67 139 16 : : PUNCT ejde-67 139 17 there there PRON ejde-67 139 18 exist exist VERB ejde-67 139 19 m̃ m̃ PROPN ejde-67 139 20 ≥ ≥ NOUN ejde-67 139 21 0 0 NUM ejde-67 139 22 and and CCONJ ejde-67 139 23 ω̃ ω̃ NUM ejde-67 139 24 ∈ ∈ NOUN ejde-67 139 25 r r NOUN ejde-67 139 26 such such ADJ ejde-67 139 27 that that PRON ejde-67 139 28 ( ( PUNCT ejde-67 139 29 ω̃,+∞ ω̃,+∞ NUM ejde-67 139 30 ) ) PUNCT ejde-67 139 31 ⊂ ⊂ PUNCT ejde-67 139 32 ρ(ã ρ(ã PROPN ejde-67 139 33 ) ) PUNCT ejde-67 139 34 and and CCONJ ejde-67 139 35 ‖(λ ‖(λ VERB ejde-67 140 1 i i VERB ejde-67 140 2 d d PROPN ejde-67 140 3 − − PROPN ejde-67 140 4 ã)−n‖ ã)−n‖ PROPN ejde-67 140 5 ≤ ≤ ADJ ejde-67 140 6 m̃ m̃ PROPN ejde-67 140 7 ( ( PUNCT ejde-67 140 8 λ− λ− PROPN ejde-67 140 9 ω̃)n ω̃)n PROPN ejde-67 140 10 for for ADP ejde-67 140 11 n n PRON ejde-67 140 12 ∈ ∈ PROPN ejde-67 140 13 n n NOUN ejde-67 140 14 and and CCONJ ejde-67 140 15 λ λ PROPN ejde-67 140 16 > > X ejde-67 140 17 ω̃. ω̃. PROPN ejde-67 140 18 theorem theorem VERB ejde-67 140 19 2.12 2.12 NUM ejde-67 140 20 ( ( PUNCT ejde-67 140 21 [ [ X ejde-67 140 22 11 11 NUM ejde-67 140 23 ] ] NUM ejde-67 140 24 ) ) PUNCT ejde-67 140 25 . . PUNCT ejde-67 141 1 assume assume VERB ejde-67 141 2 that that SCONJ ejde-67 141 3 ( ( PUNCT ejde-67 141 4 h1 h1 NOUN ejde-67 141 5 ) ) PUNCT ejde-67 141 6 holds hold VERB ejde-67 141 7 . . PUNCT ejde-67 142 1 then then ADV ejde-67 142 2 , , PUNCT ejde-67 142 3 for for ADP ejde-67 142 4 all all DET ejde-67 142 5 ϕ ϕ NOUN ejde-67 142 6 ∈ ∈ PROPN ejde-67 142 7 c c NOUN ejde-67 142 8 the the DET ejde-67 142 9 solution solution NOUN ejde-67 142 10 x x X ejde-67 142 11 of of ADP ejde-67 142 12 ( ( PUNCT ejde-67 142 13 1.3 1.3 NUM ejde-67 142 14 ) ) PUNCT ejde-67 142 15 is be AUX ejde-67 142 16 given give VERB ejde-67 142 17 by by ADP ejde-67 142 18 the the DET ejde-67 142 19 variation variation NOUN ejde-67 142 20 of of ADP ejde-67 142 21 constants constant NOUN ejde-67 142 22 formula formula NOUN ejde-67 142 23 xt xt PUNCT ejde-67 143 1 = = SYM ejde-67 143 2 v v X ejde-67 143 3 ( ( PUNCT ejde-67 143 4 t)ϕ+ t)ϕ+ PROPN ejde-67 143 5 lim lim PROPN ejde-67 143 6 λ→+∞ λ→+∞ PROPN ejde-67 143 7 ∫ ∫ PROPN ejde-67 143 8 t t PROPN ejde-67 143 9 0 0 NUM ejde-67 143 10 v v NOUN ejde-67 143 11 ( ( PUNCT ejde-67 143 12 t− t− PROPN ejde-67 143 13 s)b̃λ(x0f(s))ds s)b̃λ(x0f(s))ds PROPN ejde-67 143 14 for for ADP ejde-67 143 15 t t PROPN ejde-67 143 16 ≥ ≥ NOUN ejde-67 143 17 0 0 NUM ejde-67 143 18 , , PUNCT ejde-67 143 19 where where SCONJ ejde-67 143 20 b̃λ b̃λ ADJ ejde-67 143 21 = = PRON ejde-67 143 22 λ(λid λ(λid NOUN ejde-67 143 23 − − NOUN ejde-67 143 24 ãv ãv NOUN ejde-67 143 25 ) ) PUNCT ejde-67 143 26 −1 −1 NOUN ejde-67 143 27 for for ADP ejde-67 143 28 λ λ PROPN ejde-67 143 29 > > X ejde-67 143 30 ω̃. ω̃. PROPN ejde-67 143 31 using use VERB ejde-67 143 32 the the DET ejde-67 143 33 quasi quasi NOUN ejde-67 143 34 - - NOUN ejde-67 143 35 compactness compactness NOUN ejde-67 143 36 of of ADP ejde-67 143 37 the the DET ejde-67 143 38 semigroup semigroup NOUN ejde-67 143 39 ( ( PUNCT ejde-67 143 40 v v NOUN ejde-67 143 41 ( ( PUNCT ejde-67 143 42 t))t≥0 t))t≥0 PROPN ejde-67 143 43 , , PUNCT ejde-67 143 44 we we PRON ejde-67 143 45 obtain obtain VERB ejde-67 143 46 the the DET ejde-67 143 47 following follow VERB ejde-67 143 48 spectral spectral ADJ ejde-67 143 49 decomposition decomposition NOUN ejde-67 143 50 result result NOUN ejde-67 143 51 . . PUNCT ejde-67 144 1 theorem theorem NOUN ejde-67 144 2 2.13 2.13 NUM ejde-67 144 3 . . PUNCT ejde-67 145 1 assume assume VERB ejde-67 145 2 that that SCONJ ejde-67 145 3 the the DET ejde-67 145 4 semigroup semigroup NOUN ejde-67 145 5 ( ( PUNCT ejde-67 145 6 t t PROPN ejde-67 145 7 ( ( PUNCT ejde-67 145 8 t))t≥0 t))t≥0 PROPN ejde-67 145 9 is be AUX ejde-67 145 10 exponentially exponentially ADV ejde-67 145 11 stable stable ADJ ejde-67 145 12 and and CCONJ ejde-67 145 13 that that SCONJ ejde-67 145 14 the the DET ejde-67 145 15 operator operator NOUN ejde-67 145 16 t t NOUN ejde-67 145 17 ( ( PUNCT ejde-67 145 18 t)l t)l NOUN ejde-67 145 19 : : PUNCT ejde-67 145 20 c c X ejde-67 145 21 → → PUNCT ejde-67 145 22 x x X ejde-67 145 23 is be AUX ejde-67 145 24 compact compact ADJ ejde-67 145 25 for for ADP ejde-67 145 26 t t PROPN ejde-67 145 27 > > X ejde-67 145 28 0 0 PROPN ejde-67 145 29 . . PUNCT ejde-67 146 1 then then ADV ejde-67 146 2 , , PUNCT ejde-67 146 3 the the DET ejde-67 146 4 space space NOUN ejde-67 146 5 c c NOUN ejde-67 146 6 is be AUX ejde-67 146 7 decomposed decompose VERB ejde-67 146 8 as as ADP ejde-67 146 9 : : PUNCT ejde-67 146 10 c c NOUN ejde-67 146 11 = = SYM ejde-67 146 12 s s PROPN ejde-67 146 13 ⊕ ⊕ PROPN ejde-67 146 14 v v NOUN ejde-67 146 15 , , PUNCT ejde-67 146 16 where where SCONJ ejde-67 146 17 s s NOUN ejde-67 146 18 and and CCONJ ejde-67 146 19 v v NOUN ejde-67 146 20 are be AUX ejde-67 146 21 spaces space NOUN ejde-67 146 22 invariant invariant ADJ ejde-67 146 23 under under ADP ejde-67 146 24 v v PROPN ejde-67 146 25 ( ( PUNCT ejde-67 146 26 t t PROPN ejde-67 146 27 ) ) PUNCT ejde-67 146 28 and and CCONJ ejde-67 146 29 there there PRON ejde-67 146 30 are be VERB ejde-67 146 31 constants constant NOUN ejde-67 147 1 α α NOUN ejde-67 147 2 > > X ejde-67 147 3 0 0 PUNCT ejde-67 147 4 and and CCONJ ejde-67 147 5 m m PROPN ejde-67 147 6 ≥ ≥ NOUN ejde-67 147 7 1 1 NUM ejde-67 147 8 such such ADJ ejde-67 147 9 that that DET ejde-67 147 10 ‖v ‖v NOUN ejde-67 147 11 ( ( PUNCT ejde-67 147 12 t)ϕ‖c t)ϕ‖c PUNCT ejde-67 147 13 ≤me−αt‖ϕ‖c ≤me−αt‖ϕ‖c ADV ejde-67 147 14 , , PUNCT ejde-67 147 15 for for ADP ejde-67 147 16 each each DET ejde-67 147 17 t t PROPN ejde-67 147 18 ≥ ≥ NOUN ejde-67 147 19 0 0 NUM ejde-67 147 20 and and CCONJ ejde-67 147 21 ϕ ϕ PROPN ejde-67 147 22 ∈ ∈ PROPN ejde-67 147 23 s. s. PROPN ejde-67 147 24 v v PROPN ejde-67 147 25 is be AUX ejde-67 147 26 a a DET ejde-67 147 27 finite finite ADJ ejde-67 147 28 dimensional dimensional ADJ ejde-67 147 29 space space NOUN ejde-67 147 30 and and CCONJ ejde-67 147 31 the the DET ejde-67 147 32 restriction restriction NOUN ejde-67 147 33 v v ADP ejde-67 147 34 ( ( PUNCT ejde-67 147 35 t t PROPN ejde-67 147 36 ) ) PUNCT ejde-67 147 37 on on ADP ejde-67 147 38 v v NUM ejde-67 147 39 is be AUX ejde-67 147 40 a a DET ejde-67 147 41 groupe groupe NOUN ejde-67 147 42 . . PUNCT ejde-67 148 1 6 6 NUM ejde-67 148 2 m. m. NOUN ejde-67 148 3 el el PROPN ejde-67 148 4 attaouy attaouy PROPN ejde-67 148 5 , , PUNCT ejde-67 148 6 k. k. PROPN ejde-67 148 7 ezzinbi ezzinbi PROPN ejde-67 148 8 , , PUNCT ejde-67 148 9 g. g. PROPN ejde-67 148 10 m. m. PROPN ejde-67 148 11 n’guérékata n’guérékata PROPN ejde-67 148 12 ejde-2023/39 ejde-2023/39 VERB ejde-67 148 13 3 3 NUM ejde-67 148 14 . . NOUN ejde-67 148 15 reduction reduction NOUN ejde-67 148 16 principle principle NOUN ejde-67 148 17 if if SCONJ ejde-67 148 18 the the DET ejde-67 148 19 semigroup semigroup NOUN ejde-67 148 20 ( ( PUNCT ejde-67 148 21 v v NOUN ejde-67 148 22 ( ( PUNCT ejde-67 148 23 t))t≥0 t))t≥0 PROPN ejde-67 148 24 is be AUX ejde-67 148 25 quasi quasi ADJ ejde-67 148 26 - - ADJ ejde-67 148 27 compact compact ADJ ejde-67 148 28 , , PUNCT ejde-67 148 29 we we PRON ejde-67 148 30 can can AUX ejde-67 148 31 apply apply VERB ejde-67 148 32 the the DET ejde-67 148 33 properties property NOUN ejde-67 148 34 and and CCONJ ejde-67 148 35 notation notation NOUN ejde-67 148 36 introduced introduce VERB ejde-67 148 37 in in ADP ejde-67 148 38 theorem theorem NOUN ejde-67 148 39 2.13 2.13 NUM ejde-67 148 40 . . PUNCT ejde-67 149 1 in in ADP ejde-67 149 2 particular particular ADJ ejde-67 149 3 , , PUNCT ejde-67 149 4 we we PRON ejde-67 149 5 set set VERB ejde-67 149 6 v v ADP ejde-67 149 7 a a DET ejde-67 149 8 space space NOUN ejde-67 149 9 of of ADP ejde-67 149 10 finite finite ADJ ejde-67 149 11 dimension dimension NOUN ejde-67 149 12 d d NOUN ejde-67 149 13 with with ADP ejde-67 149 14 a a DET ejde-67 149 15 vector vector NOUN ejde-67 149 16 basis basis NOUN ejde-67 149 17 φ φ NOUN ejde-67 149 18 = = SYM ejde-67 149 19 { { PUNCT ejde-67 149 20 φ1 φ1 PROPN ejde-67 149 21 , , PUNCT ejde-67 149 22 φ2 φ2 PROPN ejde-67 149 23 . . PUNCT ejde-67 149 24 . . PUNCT ejde-67 149 25 . . PUNCT ejde-67 150 1 , , PUNCT ejde-67 150 2 φd φd AUX ejde-67 150 3 } } PUNCT ejde-67 150 4 . . PUNCT ejde-67 151 1 then then ADV ejde-67 151 2 there there PRON ejde-67 151 3 exist exist VERB ejde-67 151 4 d d NOUN ejde-67 151 5 - - PUNCT ejde-67 151 6 elements element NOUN ejde-67 151 7 ( ( PUNCT ejde-67 151 8 ψ1 ψ1 NOUN ejde-67 151 9 , , PUNCT ejde-67 151 10 ψ2 ψ2 NOUN ejde-67 151 11 , , PUNCT ejde-67 151 12 . . PUNCT ejde-67 151 13 . . PUNCT ejde-67 151 14 . . PUNCT ejde-67 152 1 , , PUNCT ejde-67 153 1 ψd ψd ADP ejde-67 153 2 ) ) PUNCT ejde-67 153 3 in in ADP ejde-67 153 4 c∗ c∗ PROPN ejde-67 153 5 such such ADJ ejde-67 153 6 that that SCONJ ejde-67 153 7 〈 〈 PROPN ejde-67 153 8 ψi ψi NOUN ejde-67 153 9 , , PUNCT ejde-67 153 10 φj φj ADP ejde-67 153 11 〉 〉 PROPN ejde-67 153 12 = = SYM ejde-67 153 13 δij δij NOUN ejde-67 153 14 , , PUNCT ejde-67 153 15 〈 〈 PROPN ejde-67 153 16 ψi ψi NOUN ejde-67 153 17 , , PUNCT ejde-67 153 18 φ φ NOUN ejde-67 153 19 〉 〉 NOUN ejde-67 153 20 = = SYM ejde-67 153 21 0 0 NUM ejde-67 153 22 , , PUNCT ejde-67 153 23 ∀φ ∀φ X ejde-67 153 24 ∈ ∈ PROPN ejde-67 153 25 s s PART ejde-67 153 26 and and CCONJ ejde-67 153 27 i i PRON ejde-67 153 28 ∈ ∈ PROPN ejde-67 153 29 { { PUNCT ejde-67 153 30 1 1 NUM ejde-67 153 31 , , PUNCT ejde-67 153 32 . . PUNCT ejde-67 153 33 . . PUNCT ejde-67 153 34 . . PUNCT ejde-67 154 1 , , PUNCT ejde-67 154 2 d d X ejde-67 154 3 } } PUNCT ejde-67 154 4 , , PUNCT ejde-67 154 5 ( ( PUNCT ejde-67 154 6 3.1 3.1 NUM ejde-67 154 7 ) ) PUNCT ejde-67 154 8 where where SCONJ ejde-67 154 9 〈 〈 PROPN ejde-67 154 10 · · SYM ejde-67 154 11 , , PUNCT ejde-67 154 12 · · PUNCT ejde-67 154 13 〉 〉 PROPN ejde-67 154 14 denotes denote VERB ejde-67 154 15 the the DET ejde-67 154 16 duality duality NOUN ejde-67 154 17 pairing pair VERB ejde-67 154 18 between between ADP ejde-67 154 19 c c PROPN ejde-67 154 20 and and CCONJ ejde-67 154 21 c∗ c∗ PROPN ejde-67 154 22 , , PUNCT ejde-67 154 23 and and CCONJ ejde-67 154 24 δij δij NOUN ejde-67 155 1 = = SYM ejde-67 155 2 { { PUNCT ejde-67 155 3 1 1 NUM ejde-67 155 4 , , PUNCT ejde-67 155 5 if if SCONJ ejde-67 155 6 i i PRON ejde-67 155 7 = = SYM ejde-67 155 8 j j PROPN ejde-67 155 9 , , PUNCT ejde-67 155 10 0 0 NUM ejde-67 155 11 , , PUNCT ejde-67 155 12 if if SCONJ ejde-67 155 13 i i PRON ejde-67 155 14 6= 6= VERB ejde-67 155 15 j. j. PROPN ejde-67 155 16 let let VERB ejde-67 155 17 ψ ψ ADP ejde-67 155 18 = = NOUN ejde-67 155 19 col{ψ1 col{ψ1 NOUN ejde-67 155 20 , , PUNCT ejde-67 155 21 ψ2 ψ2 NOUN ejde-67 155 22 , , PUNCT ejde-67 155 23 . . PUNCT ejde-67 155 24 . . PUNCT ejde-67 155 25 . . PUNCT ejde-67 156 1 , , PUNCT ejde-67 156 2 ψd ψd ADP ejde-67 156 3 } } PUNCT ejde-67 156 4 , , PUNCT ejde-67 156 5 〈 〈 PROPN ejde-67 156 6 ψ ψ X ejde-67 156 7 , , PUNCT ejde-67 156 8 φ φ PROPN ejde-67 156 9 〉 〉 PROPN ejde-67 156 10 is be AUX ejde-67 156 11 a a DET ejde-67 156 12 ( ( PUNCT ejde-67 156 13 d× d× NOUN ejde-67 156 14 d)-matrix d)-matrix NOUN ejde-67 156 15 , , PUNCT ejde-67 156 16 where where SCONJ ejde-67 156 17 the the DET ejde-67 156 18 ( ( PUNCT ejde-67 156 19 i i NOUN ejde-67 156 20 , , PUNCT ejde-67 156 21 j)-component j)-component PROPN ejde-67 156 22 is be AUX ejde-67 156 23 〈 〈 PROPN ejde-67 156 24 ψi ψi NOUN ejde-67 156 25 , , PUNCT ejde-67 156 26 ϕi ϕi ADP ejde-67 156 27 〉 〉 NOUN ejde-67 156 28 and and CCONJ ejde-67 156 29 denote denote VERB ejde-67 156 30 by by ADP ejde-67 156 31 πv πv PROPN ejde-67 156 32 and and CCONJ ejde-67 156 33 πs πs ADP ejde-67 156 34 the the DET ejde-67 156 35 projections projection NOUN ejde-67 156 36 respectively respectively ADV ejde-67 156 37 on on ADP ejde-67 156 38 v v NOUN ejde-67 156 39 and and CCONJ ejde-67 156 40 s. s. PROPN ejde-67 156 41 for for ADP ejde-67 156 42 each each DET ejde-67 156 43 ϕ ϕ PROPN ejde-67 156 44 ∈ ∈ PROPN ejde-67 157 1 c c X ejde-67 157 2 we we PRON ejde-67 157 3 have have VERB ejde-67 157 4 πvϕ πvϕ NOUN ejde-67 157 5 = = SYM ejde-67 157 6 φ〈ψ φ〈ψ X ejde-67 157 7 , , PUNCT ejde-67 157 8 ϕ ϕ PROPN ejde-67 157 9 〉 〉 PROPN ejde-67 157 10 . . PUNCT ejde-67 158 1 in in ADP ejde-67 158 2 fact fact NOUN ejde-67 158 3 , , PUNCT ejde-67 158 4 for for ADP ejde-67 158 5 ϕ ϕ PROPN ejde-67 158 6 ∈ ∈ PROPN ejde-67 158 7 c c X ejde-67 158 8 , , PUNCT ejde-67 158 9 we we PRON ejde-67 158 10 have have VERB ejde-67 158 11 ϕ ϕ NOUN ejde-67 158 12 = = NOUN ejde-67 158 13 πvϕ πvϕ PRON ejde-67 158 14 + + X ejde-67 158 15 πsϕ πsϕ X ejde-67 158 16 with with ADP ejde-67 158 17 πvϕ πvϕ NOUN ejde-67 158 18 = = PUNCT ejde-67 158 19 ∑d ∑d PROPN ejde-67 159 1 i=1 i=1 PROPN ejde-67 160 1 αiφi αiφi PROPN ejde-67 160 2 and and CCONJ ejde-67 160 3 αi αi PROPN ejde-67 160 4 ∈ ∈ PROPN ejde-67 160 5 r. r. PROPN ejde-67 160 6 from from ADP ejde-67 160 7 ( ( PUNCT ejde-67 160 8 3.1 3.1 NUM ejde-67 160 9 ) ) PUNCT ejde-67 160 10 we we PRON ejde-67 160 11 conclude conclude VERB ejde-67 160 12 that that PRON ejde-67 160 13 αi αi ADV ejde-67 160 14 = = SYM ejde-67 160 15 〈 〈 PROPN ejde-67 160 16 ψi ψi NOUN ejde-67 160 17 , , PUNCT ejde-67 160 18 ϕ ϕ PROPN ejde-67 160 19 〉 〉 NOUN ejde-67 160 20 . . PUNCT ejde-67 161 1 hence hence ADV ejde-67 161 2 πvϕ πvϕ ADV ejde-67 161 3 = = PUNCT ejde-67 161 4 d∑ d∑ PROPN ejde-67 161 5 i=1 i=1 PROPN ejde-67 162 1 〈 〈 PROPN ejde-67 162 2 ψi ψi NOUN ejde-67 162 3 , , PUNCT ejde-67 162 4 ϕ〉φi ϕ〉φi PROPN ejde-67 162 5 = = PRON ejde-67 162 6 φ〈ψ φ〈ψ PROPN ejde-67 162 7 , , PUNCT ejde-67 162 8 ϕ ϕ PROPN ejde-67 162 9 〉 〉 PROPN ejde-67 162 10 . . PUNCT ejde-67 163 1 since since SCONJ ejde-67 163 2 v v ADP ejde-67 163 3 v(t v(t NOUN ejde-67 163 4 ) ) PUNCT ejde-67 163 5 is be AUX ejde-67 163 6 a a DET ejde-67 163 7 group group NOUN ejde-67 163 8 on on ADP ejde-67 163 9 v v NOUN ejde-67 163 10 , , PUNCT ejde-67 163 11 then then ADV ejde-67 163 12 there there PRON ejde-67 163 13 exists exist VERB ejde-67 163 14 a a PRON ejde-67 163 15 ( ( PUNCT ejde-67 163 16 d×d)-matrix d×d)-matrix PROPN ejde-67 163 17 g g PROPN ejde-67 163 18 such such ADJ ejde-67 163 19 that that PRON ejde-67 163 20 v v ADP ejde-67 163 21 v(t)φ v(t)φ PROPN ejde-67 163 22 = = PUNCT ejde-67 163 23 φegt φegt VERB ejde-67 163 24 , , PUNCT ejde-67 163 25 for for ADP ejde-67 163 26 t t PROPN ejde-67 163 27 ≥ ≥ NOUN ejde-67 163 28 0 0 NUM ejde-67 163 29 . . PUNCT ejde-67 164 1 moreover moreover ADV ejde-67 164 2 , , PUNCT ejde-67 164 3 for for ADP ejde-67 164 4 each each DET ejde-67 164 5 n n CCONJ ejde-67 164 6 , , PUNCT ejde-67 164 7 n0 n0 NUM ejde-67 164 8 ∈ ∈ PROPN ejde-67 164 9 n n CCONJ ejde-67 164 10 such such ADJ ejde-67 164 11 that that SCONJ ejde-67 164 12 n n NUM ejde-67 164 13 ≥ ≥ NOUN ejde-67 164 14 n0 n0 NUM ejde-67 164 15 ≥ ≥ PROPN ejde-67 164 16 ω̃ ω̃ NUM ejde-67 164 17 and and CCONJ ejde-67 164 18 i i NOUN ejde-67 164 19 ∈ ∈ PROPN ejde-67 164 20 { { PUNCT ejde-67 164 21 1 1 NUM ejde-67 164 22 , , PUNCT ejde-67 164 23 . . PUNCT ejde-67 164 24 . . PUNCT ejde-67 164 25 . . PUNCT ejde-67 165 1 , , PUNCT ejde-67 165 2 d d X ejde-67 165 3 } } PUNCT ejde-67 165 4 , , PUNCT ejde-67 165 5 we we PRON ejde-67 165 6 define define VERB ejde-67 165 7 the the DET ejde-67 165 8 linear linear ADJ ejde-67 165 9 operator operator NOUN ejde-67 165 10 x∗i x∗i NUM ejde-67 165 11 , , PUNCT ejde-67 165 12 n n CCONJ ejde-67 165 13 by by ADP ejde-67 165 14 x∗i x∗i NUM ejde-67 165 15 , , PUNCT ejde-67 165 16 n(a n(a PRON ejde-67 165 17 ) ) PUNCT ejde-67 166 1 = = PUNCT ejde-67 167 1 〈 〈 PROPN ejde-67 167 2 ψi ψi NOUN ejde-67 167 3 , , PUNCT ejde-67 167 4 b̃nx0a b̃nx0a PROPN ejde-67 167 5 〉 〉 PROPN ejde-67 167 6 , , PUNCT ejde-67 167 7 for for ADP ejde-67 167 8 a a DET ejde-67 167 9 ∈ ∈ PROPN ejde-67 167 10 x. x. NOUN ejde-67 167 11 since since SCONJ ejde-67 167 12 |b̃n| |b̃n| VERB ejde-67 167 13 ≤ ≤ PROPN ejde-67 167 14 n n PRON ejde-67 167 15 n−ω̃m̃ n−ω̃m̃ NUM ejde-67 167 16 for for ADP ejde-67 167 17 any any DET ejde-67 167 18 n n PRON ejde-67 167 19 ≥ ≥ NOUN ejde-67 167 20 n0 n0 NUM ejde-67 167 21 , , PUNCT ejde-67 167 22 then then ADV ejde-67 167 23 x∗i x∗i NUM ejde-67 167 24 , , PUNCT ejde-67 167 25 n n PRON ejde-67 167 26 is be AUX ejde-67 167 27 a a DET ejde-67 167 28 bounded bounded ADJ ejde-67 167 29 linear linear ADJ ejde-67 167 30 operator operator NOUN ejde-67 167 31 from from ADP ejde-67 167 32 x x PUNCT ejde-67 167 33 to to ADP ejde-67 167 34 r r NOUN ejde-67 167 35 such such ADJ ejde-67 167 36 that that DET ejde-67 167 37 |x∗i |x∗i ADJ ejde-67 167 38 , , PUNCT ejde-67 167 39 n| n| NOUN ejde-67 167 40 ≤ ≤ NOUN ejde-67 167 41 n n PRON ejde-67 167 42 n− n− PROPN ejde-67 167 43 n0 n0 NOUN ejde-67 167 44 m̃ m̃ PROPN ejde-67 167 45 |ψi| |ψi| PROPN ejde-67 167 46 for for ADP ejde-67 167 47 all all PRON ejde-67 167 48 n n PRON ejde-67 167 49 ≥ ≥ NOUN ejde-67 167 50 n0 n0 NUM ejde-67 167 51 . . PUNCT ejde-67 168 1 define define VERB ejde-67 168 2 the the DET ejde-67 168 3 d d ADJ ejde-67 168 4 - - PUNCT ejde-67 168 5 column column NOUN ejde-67 168 6 vector vector NOUN ejde-67 168 7 x∗n x∗n PUNCT ejde-67 168 8 = = SYM ejde-67 168 9 col(x∗1,n col(x∗1,n PROPN ejde-67 168 10 , , PUNCT ejde-67 168 11 . . PUNCT ejde-67 168 12 . . PUNCT ejde-67 168 13 . . PUNCT ejde-67 169 1 , , PUNCT ejde-67 169 2 x x X ejde-67 169 3 ∗ ∗ NOUN ejde-67 169 4 d d PROPN ejde-67 169 5 , , PUNCT ejde-67 169 6 n n CCONJ ejde-67 169 7 ) ) PUNCT ejde-67 169 8 then then ADV ejde-67 169 9 〈 〈 PROPN ejde-67 169 10 x∗n x∗n PROPN ejde-67 169 11 , , PUNCT ejde-67 169 12 a a DET ejde-67 169 13 〉 〉 NOUN ejde-67 169 14 = = SYM ejde-67 169 15 〈 〈 PROPN ejde-67 169 16 ψ ψ X ejde-67 169 17 , , PUNCT ejde-67 169 18 b̃nx0a b̃nx0a PROPN ejde-67 169 19 〉 〉 PROPN ejde-67 169 20 , , PUNCT ejde-67 169 21 ∀a ∀a NOUN ejde-67 169 22 ∈ ∈ NOUN ejde-67 169 23 x x NOUN ejde-67 169 24 , , PUNCT ejde-67 169 25 with with ADP ejde-67 169 26 〈 〈 PROPN ejde-67 169 27 x∗n x∗n X ejde-67 169 28 , , PUNCT ejde-67 169 29 a〉i a〉i NOUN ejde-67 169 30 = = SYM ejde-67 169 31 〈 〈 PROPN ejde-67 169 32 ψi ψi NOUN ejde-67 169 33 , , PUNCT ejde-67 169 34 b̃nx0a b̃nx0a PROPN ejde-67 169 35 〉 〉 PROPN ejde-67 169 36 , , PUNCT ejde-67 169 37 for for ADP ejde-67 169 38 i i PROPN ejde-67 169 39 = = NOUN ejde-67 169 40 1 1 NUM ejde-67 169 41 , , PUNCT ejde-67 169 42 . . PUNCT ejde-67 169 43 . . PUNCT ejde-67 169 44 . . PUNCT ejde-67 170 1 , , PUNCT ejde-67 170 2 d d NOUN ejde-67 170 3 and and CCONJ ejde-67 170 4 a a DET ejde-67 170 5 ∈ ∈ NOUN ejde-67 170 6 x. x. NOUN ejde-67 170 7 consequently consequently ADV ejde-67 170 8 , , PUNCT ejde-67 170 9 supn≥n0 supn≥n0 ADJ ejde-67 170 10 |x∗n| |x∗n| PROPN ejde-67 170 11 < < X ejde-67 170 12 ∞ ∞ PROPN ejde-67 170 13 , , PUNCT ejde-67 170 14 which which PRON ejde-67 170 15 implies imply VERB ejde-67 170 16 that that SCONJ ejde-67 170 17 ( ( PUNCT ejde-67 170 18 x∗n)n≥n0 x∗n)n≥n0 NOUN ejde-67 170 19 is be AUX ejde-67 170 20 a a DET ejde-67 170 21 bounded bounded ADJ ejde-67 170 22 sequence sequence NOUN ejde-67 170 23 in in ADP ejde-67 170 24 l(x l(x PROPN ejde-67 170 25 , , PUNCT ejde-67 170 26 rd rd NOUN ejde-67 170 27 ) ) PUNCT ejde-67 170 28 . . PUNCT ejde-67 171 1 as as ADP ejde-67 171 2 a a DET ejde-67 171 3 result result NOUN ejde-67 171 4 , , PUNCT ejde-67 171 5 we we PRON ejde-67 171 6 obtain obtain VERB ejde-67 171 7 the the DET ejde-67 171 8 following follow VERB ejde-67 171 9 important important ADJ ejde-67 171 10 result result NOUN ejde-67 171 11 . . PUNCT ejde-67 172 1 theorem theorem VERB ejde-67 172 2 3.1 3.1 NUM ejde-67 172 3 ( ( PUNCT ejde-67 172 4 [ [ X ejde-67 172 5 11 11 NUM ejde-67 172 6 ] ] NUM ejde-67 172 7 ) ) PUNCT ejde-67 172 8 . . PUNCT ejde-67 173 1 there there PRON ejde-67 173 2 exists exist VERB ejde-67 173 3 x∗ x∗ PROPN ejde-67 173 4 ∈ ∈ PROPN ejde-67 173 5 l(x l(x PROPN ejde-67 173 6 , , PUNCT ejde-67 173 7 rd rd PROPN ejde-67 173 8 ) ) PUNCT ejde-67 173 9 , , PUNCT ejde-67 173 10 such such ADJ ejde-67 173 11 that that SCONJ ejde-67 173 12 ( ( PUNCT ejde-67 173 13 x∗n)n≥n0 x∗n)n≥n0 NOUN ejde-67 173 14 converges converge VERB ejde-67 173 15 weakly weakly ADV ejde-67 173 16 to to ADP ejde-67 173 17 x∗ x∗ PROPN ejde-67 173 18 : : PUNCT ejde-67 173 19 〈 〈 PROPN ejde-67 173 20 x∗n x∗n X ejde-67 173 21 , , PUNCT ejde-67 173 22 x x PROPN ejde-67 173 23 〉 〉 NUM ejde-67 173 24 → → SYM ejde-67 173 25 n→∞ n→∞ NUM ejde-67 173 26 〈 〈 NOUN ejde-67 173 27 x∗ x∗ PROPN ejde-67 173 28 , , PUNCT ejde-67 173 29 x x NOUN ejde-67 173 30 〉 〉 NOUN ejde-67 173 31 for for ADP ejde-67 173 32 all all DET ejde-67 173 33 x x SYM ejde-67 173 34 ∈ ∈ ADJ ejde-67 173 35 x. x. NOUN ejde-67 173 36 as as ADP ejde-67 173 37 a a DET ejde-67 173 38 consequence consequence NOUN ejde-67 173 39 , , PUNCT ejde-67 173 40 we we PRON ejde-67 173 41 conclude conclude VERB ejde-67 173 42 that that PRON ejde-67 173 43 corollary corollary ADJ ejde-67 173 44 3.2 3.2 NUM ejde-67 173 45 ( ( PUNCT ejde-67 173 46 [ [ X ejde-67 173 47 11 11 NUM ejde-67 173 48 ] ] NUM ejde-67 173 49 ) ) PUNCT ejde-67 173 50 . . PUNCT ejde-67 174 1 for for ADP ejde-67 174 2 any any DET ejde-67 174 3 continuous continuous ADJ ejde-67 174 4 function function NOUN ejde-67 174 5 h h NOUN ejde-67 174 6 : : PUNCT ejde-67 174 7 r→ r→ PROPN ejde-67 174 8 x x X ejde-67 174 9 , , PUNCT ejde-67 174 10 we we PRON ejde-67 174 11 have have VERB ejde-67 174 12 lim lim PROPN ejde-67 174 13 n→∞ n→∞ NUM ejde-67 175 1 ∫ ∫ PROPN ejde-67 175 2 t t PROPN ejde-67 175 3 σ σ PROPN ejde-67 175 4 v v PROPN ejde-67 175 5 v(t− v(t− PROPN ejde-67 175 6 ξ)πv(b̃nx0h(ξ ξ)πv(b̃nx0h(ξ NOUN ejde-67 175 7 ) ) PUNCT ejde-67 175 8 ) ) PUNCT ejde-67 176 1 dξ dξ PROPN ejde-67 176 2 = = PUNCT ejde-67 177 1 φ φ PROPN ejde-67 177 2 ∫ ∫ PROPN ejde-67 177 3 t t PROPN ejde-67 177 4 σ σ PROPN ejde-67 177 5 e(t−ξ)g〈x∗ e(t−ξ)g〈x∗ PROPN ejde-67 177 6 , , PUNCT ejde-67 177 7 h(ξ h(ξ PROPN ejde-67 177 8 ) ) PUNCT ejde-67 177 9 〉 〉 PROPN ejde-67 177 10 dξ dξ PROPN ejde-67 177 11 , , PUNCT ejde-67 177 12 for for ADP ejde-67 177 13 t t PROPN ejde-67 177 14 , , PUNCT ejde-67 177 15 σ σ PROPN ejde-67 177 16 ∈ ∈ PROPN ejde-67 177 17 r. r. PROPN ejde-67 177 18 ejde-2023/39 ejde-2023/39 PROPN ejde-67 177 19 reduction reduction NOUN ejde-67 177 20 principle principle NOUN ejde-67 177 21 7 7 NUM ejde-67 177 22 theorem theorem VERB ejde-67 177 23 3.3 3.3 NUM ejde-67 177 24 ( ( PUNCT ejde-67 177 25 [ [ X ejde-67 177 26 11 11 NUM ejde-67 177 27 ] ] NUM ejde-67 177 28 ) ) PUNCT ejde-67 177 29 . . PUNCT ejde-67 178 1 assume assume VERB ejde-67 178 2 that that SCONJ ejde-67 178 3 the the DET ejde-67 178 4 semigroup semigroup NOUN ejde-67 178 5 ( ( PUNCT ejde-67 178 6 t t PROPN ejde-67 178 7 ( ( PUNCT ejde-67 178 8 t))t≥0 t))t≥0 PROPN ejde-67 178 9 is be AUX ejde-67 178 10 compact compact ADJ ejde-67 178 11 . . PUNCT ejde-67 179 1 moreover moreover ADV ejde-67 179 2 , , PUNCT ejde-67 179 3 f f PROPN ejde-67 179 4 is be AUX ejde-67 179 5 continuous continuous ADJ ejde-67 179 6 and and CCONJ ejde-67 179 7 x x PRON ejde-67 179 8 is be AUX ejde-67 179 9 a a DET ejde-67 179 10 solution solution NOUN ejde-67 179 11 of of ADP ejde-67 179 12 ( ( PUNCT ejde-67 179 13 1.3 1.3 NUM ejde-67 179 14 ) ) PUNCT ejde-67 179 15 on on ADP ejde-67 179 16 r r NOUN ejde-67 179 17 , , PUNCT ejde-67 179 18 then then ADV ejde-67 179 19 z(t z(t NOUN ejde-67 179 20 ) ) PUNCT ejde-67 179 21 = = PUNCT ejde-67 180 1 〈 〈 PROPN ejde-67 180 2 ψ ψ X ejde-67 180 3 , , PUNCT ejde-67 180 4 xt xt PROPN ejde-67 180 5 〉 〉 PROPN ejde-67 180 6 is be AUX ejde-67 180 7 a a DET ejde-67 180 8 solution solution NOUN ejde-67 180 9 of of ADP ejde-67 180 10 the the DET ejde-67 180 11 ordinary ordinary ADJ ejde-67 180 12 differential differential ADJ ejde-67 180 13 equation equation NOUN ejde-67 180 14 z′(t z′(t NOUN ejde-67 180 15 ) ) PUNCT ejde-67 180 16 = = PUNCT ejde-67 180 17 gz(t gz(t X ejde-67 180 18 ) ) PUNCT ejde-67 181 1 + + CCONJ ejde-67 181 2 〈 〈 PROPN ejde-67 181 3 x∗ x∗ NOUN ejde-67 181 4 , , PUNCT ejde-67 181 5 f(t f(t PROPN ejde-67 181 6 ) ) PUNCT ejde-67 181 7 〉 〉 PROPN ejde-67 181 8 , , PUNCT ejde-67 181 9 for for ADP ejde-67 181 10 t t PROPN ejde-67 181 11 ∈ ∈ PROPN ejde-67 181 12 r. r. PROPN ejde-67 181 13 ( ( PUNCT ejde-67 181 14 3.2 3.2 NUM ejde-67 181 15 ) ) PUNCT ejde-67 181 16 conversely conversely ADV ejde-67 181 17 , , PUNCT ejde-67 181 18 if if SCONJ ejde-67 181 19 z z X ejde-67 181 20 ( ( PUNCT ejde-67 181 21 · · PUNCT ejde-67 181 22 ) ) PUNCT ejde-67 181 23 is be AUX ejde-67 181 24 a a DET ejde-67 181 25 solution solution NOUN ejde-67 181 26 of of ADP ejde-67 181 27 ( ( PUNCT ejde-67 181 28 3.2 3.2 NUM ejde-67 181 29 ) ) PUNCT ejde-67 181 30 on on ADP ejde-67 181 31 r r NOUN ejde-67 181 32 and and CCONJ ejde-67 181 33 f f PROPN ejde-67 181 34 is be AUX ejde-67 181 35 bounded bound VERB ejde-67 181 36 then then ADV ejde-67 181 37 the the DET ejde-67 181 38 function function NOUN ejde-67 181 39 x(t x(t PROPN ejde-67 181 40 ) ) PUNCT ejde-67 182 1 = = PUNCT ejde-67 182 2 [ [ PUNCT ejde-67 182 3 φ φ X ejde-67 182 4 z(t z(t PROPN ejde-67 182 5 ) ) PUNCT ejde-67 183 1 + + CCONJ ejde-67 183 2 lim lim PROPN ejde-67 183 3 n→∞ n→∞ NUM ejde-67 183 4 ∫ ∫ PROPN ejde-67 183 5 t t PROPN ejde-67 183 6 −∞ −∞ ADP ejde-67 183 7 v v PROPN ejde-67 183 8 s(t− s(t− PROPN ejde-67 183 9 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 183 10 ) ) PUNCT ejde-67 183 11 ) ) PUNCT ejde-67 183 12 dξ dξ CCONJ ejde-67 183 13 ] ] PUNCT ejde-67 183 14 ( ( PUNCT ejde-67 183 15 0 0 NUM ejde-67 183 16 ) ) PUNCT ejde-67 183 17 , , PUNCT ejde-67 183 18 for for ADP ejde-67 183 19 t t PROPN ejde-67 183 20 ∈ ∈ PROPN ejde-67 183 21 r r NOUN ejde-67 183 22 , , PUNCT ejde-67 183 23 is be AUX ejde-67 183 24 a a DET ejde-67 183 25 mild mild ADJ ejde-67 183 26 solution solution NOUN ejde-67 183 27 of of ADP ejde-67 183 28 ( ( PUNCT ejde-67 183 29 1.3 1.3 NUM ejde-67 183 30 ) ) PUNCT ejde-67 183 31 on on ADP ejde-67 183 32 r. r. PROPN ejde-67 183 33 as as ADP ejde-67 183 34 a a DET ejde-67 183 35 consequence consequence NOUN ejde-67 183 36 of of ADP ejde-67 183 37 the the DET ejde-67 183 38 above above NOUN ejde-67 183 39 , , PUNCT ejde-67 183 40 we we PRON ejde-67 183 41 establish establish VERB ejde-67 183 42 the the DET ejde-67 183 43 following follow VERB ejde-67 183 44 fundamental fundamental ADJ ejde-67 183 45 reduction reduction NOUN ejde-67 183 46 principle principle NOUN ejde-67 183 47 which which PRON ejde-67 183 48 allows allow VERB ejde-67 183 49 us we PRON ejde-67 183 50 to to PART ejde-67 183 51 prove prove VERB ejde-67 183 52 the the DET ejde-67 183 53 existence existence NOUN ejde-67 183 54 of of ADP ejde-67 183 55 an an DET ejde-67 183 56 almost almost ADV ejde-67 183 57 automorphic automorphic ADJ ejde-67 183 58 and and CCONJ ejde-67 183 59 almost almost ADV ejde-67 183 60 periodic periodic ADJ ejde-67 183 61 solution solution NOUN ejde-67 183 62 of of ADP ejde-67 183 63 the the DET ejde-67 183 64 ( ( PUNCT ejde-67 183 65 1.3 1.3 NUM ejde-67 183 66 ) ) PUNCT ejde-67 183 67 . . PUNCT ejde-67 184 1 theorem theorem VERB ejde-67 184 2 3.4 3.4 NUM ejde-67 184 3 . . PUNCT ejde-67 185 1 assume assume VERB ejde-67 185 2 that that SCONJ ejde-67 185 3 the the DET ejde-67 185 4 semigroup semigroup NOUN ejde-67 185 5 ( ( PUNCT ejde-67 185 6 t t PROPN ejde-67 185 7 ( ( PUNCT ejde-67 185 8 t))t≥0 t))t≥0 PROPN ejde-67 185 9 is be AUX ejde-67 185 10 exponentially exponentially ADV ejde-67 185 11 stable stable ADJ ejde-67 185 12 and and CCONJ ejde-67 185 13 that that SCONJ ejde-67 185 14 the the DET ejde-67 185 15 operator operator NOUN ejde-67 185 16 t t PROPN ejde-67 185 17 ( ( PUNCT ejde-67 185 18 t)l t)l NOUN ejde-67 185 19 is be AUX ejde-67 185 20 compact compact ADJ ejde-67 185 21 for for ADP ejde-67 185 22 t t PROPN ejde-67 185 23 > > X ejde-67 185 24 0 0 X ejde-67 185 25 . . PUNCT ejde-67 186 1 moreover moreover ADV ejde-67 186 2 , , PUNCT ejde-67 186 3 f f PROPN ejde-67 186 4 is be AUX ejde-67 186 5 locally locally ADV ejde-67 186 6 integrable integrable ADJ ejde-67 186 7 and and CCONJ ejde-67 186 8 x x X ejde-67 186 9 is be AUX ejde-67 186 10 a a DET ejde-67 186 11 solution solution NOUN ejde-67 186 12 of of ADP ejde-67 186 13 ( ( PUNCT ejde-67 186 14 1.3 1.3 NUM ejde-67 186 15 ) ) PUNCT ejde-67 186 16 on on ADP ejde-67 186 17 r r NOUN ejde-67 186 18 , , PUNCT ejde-67 186 19 then then ADV ejde-67 186 20 z(t z(t NOUN ejde-67 186 21 ) ) PUNCT ejde-67 187 1 = = PUNCT ejde-67 187 2 〈 〈 PROPN ejde-67 187 3 ψ ψ X ejde-67 187 4 , , PUNCT ejde-67 187 5 xt xt PROPN ejde-67 187 6 〉 〉 PROPN ejde-67 187 7 is be AUX ejde-67 187 8 a a DET ejde-67 187 9 solution solution NOUN ejde-67 187 10 of of ADP ejde-67 187 11 the the DET ejde-67 187 12 ordinary ordinary ADJ ejde-67 187 13 differential differential ADJ ejde-67 187 14 equation equation NOUN ejde-67 187 15 z′(t z′(t NOUN ejde-67 187 16 ) ) PUNCT ejde-67 187 17 = = PUNCT ejde-67 187 18 gz(t gz(t X ejde-67 187 19 ) ) PUNCT ejde-67 188 1 + + CCONJ ejde-67 188 2 〈 〈 PROPN ejde-67 188 3 x∗ x∗ NOUN ejde-67 188 4 , , PUNCT ejde-67 188 5 f(t f(t PROPN ejde-67 188 6 ) ) PUNCT ejde-67 188 7 〉 〉 PROPN ejde-67 188 8 , , PUNCT ejde-67 188 9 for for ADP ejde-67 188 10 t t PROPN ejde-67 188 11 ∈ ∈ PROPN ejde-67 188 12 r. r. PROPN ejde-67 188 13 ( ( PUNCT ejde-67 188 14 3.3 3.3 NUM ejde-67 188 15 ) ) PUNCT ejde-67 188 16 conversely conversely ADV ejde-67 188 17 , , PUNCT ejde-67 188 18 if if SCONJ ejde-67 188 19 z z X ejde-67 188 20 ( ( PUNCT ejde-67 188 21 · · PUNCT ejde-67 188 22 ) ) PUNCT ejde-67 188 23 is be AUX ejde-67 188 24 a a DET ejde-67 188 25 solution solution NOUN ejde-67 188 26 of of ADP ejde-67 188 27 ( ( PUNCT ejde-67 188 28 3.3 3.3 NUM ejde-67 188 29 ) ) PUNCT ejde-67 188 30 on on ADP ejde-67 188 31 r r NOUN ejde-67 188 32 and and CCONJ ejde-67 188 33 sup sup NOUN ejde-67 188 34 t∈r t∈r NOUN ejde-67 188 35 ( ( PUNCT ejde-67 188 36 ∫ ∫ PROPN ejde-67 188 37 t+1 t+1 PROPN ejde-67 188 38 t t PROPN ejde-67 188 39 ‖f(s)‖pds ‖f(s)‖pds PROPN ejde-67 188 40 ) ) PUNCT ejde-67 189 1 1 1 X ejde-67 189 2 / / SYM ejde-67 189 3 p p X ejde-67 189 4 < < X ejde-67 189 5 ∞ ∞ PROPN ejde-67 189 6 then then ADV ejde-67 189 7 the the DET ejde-67 189 8 function function NOUN ejde-67 189 9 x(t x(t PROPN ejde-67 189 10 ) ) PUNCT ejde-67 190 1 = = PUNCT ejde-67 190 2 [ [ PUNCT ejde-67 190 3 φ φ X ejde-67 190 4 z(t z(t PROPN ejde-67 190 5 ) ) PUNCT ejde-67 191 1 + + CCONJ ejde-67 191 2 lim lim PROPN ejde-67 191 3 n→∞ n→∞ NUM ejde-67 191 4 ∫ ∫ PROPN ejde-67 191 5 t t PROPN ejde-67 191 6 −∞ −∞ ADP ejde-67 191 7 v v PROPN ejde-67 191 8 s(t− s(t− PROPN ejde-67 191 9 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 191 10 ) ) PUNCT ejde-67 191 11 ) ) PUNCT ejde-67 191 12 dξ dξ CCONJ ejde-67 191 13 ] ] PUNCT ejde-67 191 14 ( ( PUNCT ejde-67 191 15 0 0 NUM ejde-67 191 16 ) ) PUNCT ejde-67 191 17 , , PUNCT ejde-67 191 18 for for ADP ejde-67 191 19 t t PROPN ejde-67 191 20 ∈ ∈ PROPN ejde-67 191 21 r r PROPN ejde-67 191 22 , , PUNCT ejde-67 191 23 ( ( PUNCT ejde-67 191 24 3.4 3.4 NUM ejde-67 191 25 ) ) PUNCT ejde-67 191 26 is be AUX ejde-67 191 27 a a DET ejde-67 191 28 solution solution NOUN ejde-67 191 29 of of ADP ejde-67 191 30 ( ( PUNCT ejde-67 191 31 1.3 1.3 NUM ejde-67 191 32 ) ) PUNCT ejde-67 191 33 on on ADP ejde-67 191 34 r. r. PROPN ejde-67 191 35 proof proof NOUN ejde-67 191 36 . . PUNCT ejde-67 192 1 the the DET ejde-67 192 2 proof proof NOUN ejde-67 192 3 is be AUX ejde-67 192 4 similar similar ADJ ejde-67 192 5 to to ADP ejde-67 192 6 that that PRON ejde-67 192 7 of of ADP ejde-67 192 8 theorem theorem ADJ ejde-67 192 9 3.3 3.3 NUM ejde-67 192 10 . . PUNCT ejde-67 193 1 we we PRON ejde-67 193 2 only only ADV ejde-67 193 3 prove prove VERB ejde-67 193 4 that that SCONJ ejde-67 193 5 lim lim PROPN ejde-67 193 6 n→∞ n→∞ PRON ejde-67 193 7 ∫ ∫ PROPN ejde-67 193 8 t t PROPN ejde-67 193 9 −∞ −∞ ADP ejde-67 193 10 v v PROPN ejde-67 193 11 s(t− s(t− PROPN ejde-67 193 12 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 193 13 ) ) PUNCT ejde-67 193 14 ) ) PUNCT ejde-67 194 1 dξ dξ PROPN ejde-67 194 2 , , PUNCT ejde-67 194 3 exists exist VERB ejde-67 194 4 in in ADP ejde-67 194 5 c. c. NOUN ejde-67 194 6 for for ADP ejde-67 194 7 t t PROPN ejde-67 194 8 ∈ ∈ PROPN ejde-67 194 9 r r NOUN ejde-67 194 10 and and CCONJ ejde-67 194 11 for for ADP ejde-67 194 12 n n PRON ejde-67 194 13 sufficiently sufficiently ADV ejde-67 194 14 large large ADJ ejde-67 194 15 , , PUNCT ejde-67 194 16 we we PRON ejde-67 194 17 have have VERB ejde-67 194 18 ‖ ‖ PROPN ejde-67 194 19 lim lim PROPN ejde-67 194 20 n→∞ n→∞ NUM ejde-67 194 21 ∫ ∫ PROPN ejde-67 194 22 t t PROPN ejde-67 194 23 −∞ −∞ ADP ejde-67 194 24 v v PROPN ejde-67 194 25 s(t− s(t− PROPN ejde-67 194 26 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 194 27 ) ) PUNCT ejde-67 194 28 ) ) PUNCT ejde-67 195 1 dξ‖ dξ‖ NOUN ejde-67 196 1 ≤ ≤ NUM ejde-67 196 2 k k NOUN ejde-67 196 3 , , PUNCT ejde-67 196 4 where where SCONJ ejde-67 196 5 k k PROPN ejde-67 196 6 = = SYM ejde-67 196 7 2m̃n‖πs‖ 2m̃n‖πs‖ NUM ejde-67 196 8 sup sup NOUN ejde-67 196 9 t∈r t∈r NOUN ejde-67 196 10 ( ( PUNCT ejde-67 196 11 ∫ ∫ PROPN ejde-67 196 12 t+1 t+1 PROPN ejde-67 196 13 t t PROPN ejde-67 196 14 ‖f(s)‖ds ‖f(s)‖ds PROPN ejde-67 196 15 ) ) PUNCT ejde-67 196 16 1 1 NUM ejde-67 196 17 1− 1− NUM ejde-67 196 18 e−α e−α VERB ejde-67 196 19 . . PUNCT ejde-67 197 1 let let VERB ejde-67 197 2 h(n h(n PROPN ejde-67 197 3 , , PUNCT ejde-67 197 4 s s PROPN ejde-67 197 5 , , PUNCT ejde-67 197 6 t t PROPN ejde-67 197 7 ) ) PUNCT ejde-67 197 8 = = PUNCT ejde-67 198 1 v v NUM ejde-67 198 2 s(t− s(t− PROPN ejde-67 198 3 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 198 4 ) ) PUNCT ejde-67 198 5 ) ) PUNCT ejde-67 198 6 , , PUNCT ejde-67 198 7 for for ADP ejde-67 198 8 n n PRON ejde-67 198 9 ∈ ∈ PROPN ejde-67 198 10 n n NOUN ejde-67 198 11 and and CCONJ ejde-67 198 12 s s VERB ejde-67 198 13 ≤ ≤ NOUN ejde-67 198 14 t. t. NOUN ejde-67 198 15 for for ADP ejde-67 198 16 n n PROPN ejde-67 198 17 and and CCONJ ejde-67 198 18 m m PRON ejde-67 198 19 sufficiently sufficiently ADV ejde-67 198 20 large large ADJ ejde-67 198 21 and and CCONJ ejde-67 198 22 σ σ PROPN ejde-67 198 23 ≤ ≤ PROPN ejde-67 198 24 t t PROPN ejde-67 198 25 , , PUNCT ejde-67 198 26 we we PRON ejde-67 198 27 have have VERB ejde-67 198 28 ‖ ‖ PROPN ejde-67 198 29 ∫ ∫ PROPN ejde-67 198 30 t t PROPN ejde-67 198 31 −∞ −∞ ADP ejde-67 198 32 h(n h(n PROPN ejde-67 198 33 , , PUNCT ejde-67 198 34 s s X ejde-67 198 35 , , PUNCT ejde-67 198 36 t)ds− t)ds− PROPN ejde-67 198 37 ∫ ∫ PROPN ejde-67 198 38 t t PROPN ejde-67 198 39 −∞ −∞ X ejde-67 198 40 h(m h(m PROPN ejde-67 198 41 , , PUNCT ejde-67 198 42 s s NOUN ejde-67 198 43 , , PUNCT ejde-67 198 44 t)ds‖ t)ds‖ NOUN ejde-67 198 45 ≤ ≤ NOUN ejde-67 198 46 2ke−α(t−σ 2ke−α(t−σ NOUN ejde-67 198 47 ) ) PUNCT ejde-67 199 1 + + CCONJ ejde-67 199 2 ‖ ‖ PROPN ejde-67 199 3 ∫ ∫ PROPN ejde-67 199 4 t t PROPN ejde-67 199 5 σ σ PROPN ejde-67 199 6 h(n h(n PROPN ejde-67 199 7 , , PUNCT ejde-67 199 8 s s X ejde-67 199 9 , , PUNCT ejde-67 199 10 t)ds− t)ds− PROPN ejde-67 199 11 ∫ ∫ PROPN ejde-67 199 12 t t PROPN ejde-67 199 13 σ σ PROPN ejde-67 199 14 h(m h(m PROPN ejde-67 199 15 , , PUNCT ejde-67 199 16 s s PROPN ejde-67 199 17 , , PUNCT ejde-67 199 18 t)ds‖. t)ds‖. ADP ejde-67 199 19 8 8 NUM ejde-67 199 20 m. m. NOUN ejde-67 199 21 el el PROPN ejde-67 199 22 attaouy attaouy PROPN ejde-67 199 23 , , PUNCT ejde-67 199 24 k. k. PROPN ejde-67 199 25 ezzinbi ezzinbi PROPN ejde-67 199 26 , , PUNCT ejde-67 199 27 g. g. PROPN ejde-67 199 28 m. m. PROPN ejde-67 199 29 n’guérékata n’guérékata PROPN ejde-67 199 30 ejde-2023/39 ejde-2023/39 PROPN ejde-67 199 31 since since SCONJ ejde-67 199 32 limn→∞ limn→∞ PROPN ejde-67 199 33 ∫ ∫ PROPN ejde-67 199 34 t t PROPN ejde-67 199 35 σ σ PROPN ejde-67 199 36 h(n h(n PROPN ejde-67 199 37 , , PUNCT ejde-67 199 38 s s X ejde-67 199 39 , , PUNCT ejde-67 199 40 t)ds t)ds PROPN ejde-67 199 41 exists exist VERB ejde-67 200 1 , , PUNCT ejde-67 200 2 it it PRON ejde-67 200 3 follows follow VERB ejde-67 200 4 that that SCONJ ejde-67 200 5 lim lim PROPN ejde-67 200 6 sup sup PROPN ejde-67 200 7 n n CCONJ ejde-67 200 8 , , PUNCT ejde-67 200 9 m→∞ m→∞ NUM ejde-67 200 10 ‖ ‖ PROPN ejde-67 200 11 ∫ ∫ PROPN ejde-67 200 12 t t PROPN ejde-67 200 13 −∞ −∞ ADP ejde-67 200 14 h(n h(n PROPN ejde-67 200 15 , , PUNCT ejde-67 200 16 s s X ejde-67 200 17 , , PUNCT ejde-67 200 18 t)ds− t)ds− PROPN ejde-67 200 19 ∫ ∫ PROPN ejde-67 200 20 t t PROPN ejde-67 200 21 −∞ −∞ X ejde-67 200 22 h(m h(m PROPN ejde-67 200 23 , , PUNCT ejde-67 200 24 s s NOUN ejde-67 200 25 , , PUNCT ejde-67 200 26 t)ds‖ t)ds‖ NOUN ejde-67 200 27 ≤ ≤ NOUN ejde-67 200 28 2ke−α(t−σ 2ke−α(t−σ NOUN ejde-67 200 29 ) ) PUNCT ejde-67 200 30 . . PUNCT ejde-67 201 1 by by ADP ejde-67 201 2 letting let VERB ejde-67 201 3 σ σ PROPN ejde-67 201 4 → → SYM ejde-67 201 5 −∞ −∞ NOUN ejde-67 201 6 , , PUNCT ejde-67 201 7 we we PRON ejde-67 201 8 obtain obtain VERB ejde-67 201 9 lim lim PROPN ejde-67 201 10 sup sup PROPN ejde-67 201 11 n n CCONJ ejde-67 201 12 , , PUNCT ejde-67 201 13 m→∞ m→∞ NUM ejde-67 201 14 ‖ ‖ PROPN ejde-67 201 15 ∫ ∫ PROPN ejde-67 201 16 t t PROPN ejde-67 201 17 −∞ −∞ ADP ejde-67 201 18 h(n h(n PROPN ejde-67 201 19 , , PUNCT ejde-67 201 20 s s X ejde-67 201 21 , , PUNCT ejde-67 201 22 t)ds− t)ds− PROPN ejde-67 201 23 ∫ ∫ PROPN ejde-67 201 24 t t PROPN ejde-67 201 25 −∞ −∞ X ejde-67 201 26 h(m h(m PROPN ejde-67 201 27 , , PUNCT ejde-67 201 28 s s NOUN ejde-67 201 29 , , PUNCT ejde-67 201 30 t)ds‖ t)ds‖ NOUN ejde-67 201 31 = = SYM ejde-67 201 32 0 0 X ejde-67 201 33 . . PUNCT ejde-67 202 1 thus thus ADV ejde-67 202 2 , , PUNCT ejde-67 202 3 by by ADP ejde-67 202 4 the the DET ejde-67 202 5 completeness completeness NOUN ejde-67 202 6 of of ADP ejde-67 202 7 the the DET ejde-67 202 8 phase phase NOUN ejde-67 202 9 space space NOUN ejde-67 202 10 c c NOUN ejde-67 202 11 , , PUNCT ejde-67 202 12 we we PRON ejde-67 202 13 deduce deduce VERB ejde-67 202 14 that that SCONJ ejde-67 202 15 lim lim PROPN ejde-67 202 16 n→∞ n→∞ PRON ejde-67 202 17 ‖ ‖ PROPN ejde-67 202 18 ∫ ∫ PROPN ejde-67 202 19 t t PROPN ejde-67 202 20 −∞ −∞ ADP ejde-67 202 21 h(n h(n PROPN ejde-67 202 22 , , PUNCT ejde-67 202 23 s s X ejde-67 202 24 , , PUNCT ejde-67 202 25 t)ds‖ t)ds‖ NOUN ejde-67 202 26 exists exist VERB ejde-67 202 27 in in ADP ejde-67 202 28 c. c. PROPN ejde-67 202 29 � � PROPN ejde-67 202 30 remark remark VERB ejde-67 202 31 3.5 3.5 NUM ejde-67 202 32 . . PUNCT ejde-67 203 1 this this DET ejde-67 203 2 principal principal ADJ ejde-67 203 3 result result NOUN ejde-67 203 4 was be AUX ejde-67 203 5 established establish VERB ejde-67 203 6 when when SCONJ ejde-67 203 7 the the DET ejde-67 203 8 semigroup semigroup NOUN ejde-67 203 9 ( ( PUNCT ejde-67 203 10 t t PROPN ejde-67 203 11 ( ( PUNCT ejde-67 203 12 t))t≥0 t))t≥0 PROPN ejde-67 203 13 is be AUX ejde-67 203 14 compact compact ADJ ejde-67 203 15 . . PUNCT ejde-67 204 1 we we PRON ejde-67 204 2 establish establish VERB ejde-67 204 3 the the DET ejde-67 204 4 same same ADJ ejde-67 204 5 result result NOUN ejde-67 204 6 even even ADV ejde-67 204 7 if if SCONJ ejde-67 204 8 the the DET ejde-67 204 9 semigroup semigroup NOUN ejde-67 204 10 is be AUX ejde-67 204 11 not not PART ejde-67 204 12 necessary necessary ADJ ejde-67 204 13 compact compact ADJ ejde-67 204 14 but but CCONJ ejde-67 204 15 the the DET ejde-67 204 16 operator operator NOUN ejde-67 204 17 t t PROPN ejde-67 204 18 ( ( PUNCT ejde-67 204 19 t)l t)l NOUN ejde-67 204 20 is be AUX ejde-67 204 21 compact compact ADJ ejde-67 204 22 for for ADP ejde-67 204 23 all all DET ejde-67 204 24 t t PROPN ejde-67 204 25 > > X ejde-67 204 26 0 0 NUM ejde-67 204 27 . . PROPN ejde-67 205 1 4 4 NUM ejde-67 205 2 . . X ejde-67 205 3 almost almost ADV ejde-67 205 4 periodicity periodicity NOUN ejde-67 205 5 and and CCONJ ejde-67 205 6 almost almost ADV ejde-67 205 7 automorphy automorphy NOUN ejde-67 205 8 in in ADP ejde-67 205 9 what what PRON ejde-67 205 10 follows follow VERB ejde-67 205 11 , , PUNCT ejde-67 205 12 we we PRON ejde-67 205 13 recall recall VERB ejde-67 205 14 some some DET ejde-67 205 15 results result NOUN ejde-67 205 16 on on ADP ejde-67 205 17 almost almost ADV ejde-67 205 18 automorphic automorphic ADJ ejde-67 205 19 functions function NOUN ejde-67 205 20 and and CCONJ ejde-67 205 21 almost almost ADV ejde-67 205 22 periodic periodic ADJ ejde-67 205 23 functions function NOUN ejde-67 205 24 . . PUNCT ejde-67 206 1 let let VERB ejde-67 206 2 bc(r bc(r NOUN ejde-67 206 3 , , PUNCT ejde-67 206 4 x x PRON ejde-67 206 5 ) ) PUNCT ejde-67 206 6 be be VERB ejde-67 206 7 the the DET ejde-67 206 8 space space NOUN ejde-67 206 9 of of ADP ejde-67 206 10 bounded bounded ADJ ejde-67 206 11 continuous continuous ADJ ejde-67 206 12 functions function NOUN ejde-67 206 13 from from ADP ejde-67 206 14 r r NOUN ejde-67 206 15 to to ADP ejde-67 206 16 x x PRON ejde-67 206 17 , , PUNCT ejde-67 206 18 provided provide VERB ejde-67 206 19 with with ADP ejde-67 206 20 the the DET ejde-67 206 21 uniform uniform ADJ ejde-67 206 22 norm norm PROPN ejde-67 206 23 topology topology NOUN ejde-67 206 24 . . PUNCT ejde-67 207 1 let let VERB ejde-67 207 2 x x PUNCT ejde-67 207 3 ∈ ∈ PROPN ejde-67 207 4 bc(r bc(r NOUN ejde-67 207 5 , , PUNCT ejde-67 207 6 x x PRON ejde-67 207 7 ) ) PUNCT ejde-67 207 8 and and CCONJ ejde-67 207 9 τ τ PROPN ejde-67 207 10 ∈ ∈ PROPN ejde-67 207 11 r r NOUN ejde-67 207 12 , , PUNCT ejde-67 207 13 we we PRON ejde-67 207 14 define define VERB ejde-67 207 15 the the DET ejde-67 207 16 translation translation NOUN ejde-67 207 17 function function VERB ejde-67 207 18 xτ xτ PROPN ejde-67 207 19 ( ( PUNCT ejde-67 207 20 s s X ejde-67 207 21 ) ) PUNCT ejde-67 207 22 = = SYM ejde-67 208 1 x(τ x(τ PROPN ejde-67 208 2 + + NUM ejde-67 208 3 s s X ejde-67 208 4 ) ) PUNCT ejde-67 208 5 for for ADP ejde-67 208 6 s s PROPN ejde-67 208 7 ∈ ∈ PROPN ejde-67 208 8 r. r. PROPN ejde-67 208 9 definition definition NOUN ejde-67 208 10 4.1 4.1 NUM ejde-67 208 11 ( ( PUNCT ejde-67 208 12 [ [ X ejde-67 208 13 6 6 NUM ejde-67 208 14 ] ] NUM ejde-67 208 15 ) ) PUNCT ejde-67 208 16 . . PUNCT ejde-67 209 1 a a DET ejde-67 209 2 bounded bounded ADJ ejde-67 209 3 continuous continuous ADJ ejde-67 209 4 function function NOUN ejde-67 209 5 x x NOUN ejde-67 209 6 : : PUNCT ejde-67 209 7 r r X ejde-67 209 8 → → PUNCT ejde-67 209 9 x x X ejde-67 209 10 is be AUX ejde-67 209 11 said say VERB ejde-67 209 12 to to PART ejde-67 209 13 be be AUX ejde-67 209 14 almost almost ADV ejde-67 209 15 periodic periodic ADJ ejde-67 209 16 if if SCONJ ejde-67 209 17 { { PUNCT ejde-67 209 18 xτ xτ PROPN ejde-67 209 19 , , PUNCT ejde-67 209 20 τ τ PROPN ejde-67 209 21 ∈ ∈ PROPN ejde-67 209 22 r r X ejde-67 209 23 } } PUNCT ejde-67 209 24 is be AUX ejde-67 209 25 relatively relatively ADV ejde-67 209 26 compact compact ADJ ejde-67 209 27 in in ADP ejde-67 209 28 bc(r bc(r NOUN ejde-67 209 29 , , PUNCT ejde-67 209 30 x x NOUN ejde-67 209 31 ) ) PUNCT ejde-67 209 32 . . PUNCT ejde-67 210 1 theorem theorem VERB ejde-67 210 2 4.2 4.2 NUM ejde-67 210 3 ( ( PUNCT ejde-67 210 4 [ [ X ejde-67 210 5 4 4 NUM ejde-67 210 6 ] ] NUM ejde-67 210 7 ) ) PUNCT ejde-67 210 8 . . PUNCT ejde-67 211 1 a a DET ejde-67 211 2 function function NOUN ejde-67 211 3 f f NOUN ejde-67 211 4 : : PUNCT ejde-67 211 5 r r X ejde-67 211 6 → → PUNCT ejde-67 211 7 x x X ejde-67 211 8 is be AUX ejde-67 211 9 almost almost ADV ejde-67 211 10 periodic periodic ADJ ejde-67 211 11 if if SCONJ ejde-67 211 12 and and CCONJ ejde-67 211 13 only only ADV ejde-67 211 14 if if SCONJ ejde-67 211 15 for for ADP ejde-67 211 16 every every DET ejde-67 211 17 sequence sequence NOUN ejde-67 211 18 of of ADP ejde-67 211 19 real real ADJ ejde-67 211 20 numbers number NOUN ejde-67 211 21 ( ( PUNCT ejde-67 211 22 s′n)n s′n)n NUM ejde-67 211 23 , , PUNCT ejde-67 211 24 there there PRON ejde-67 211 25 exist exist VERB ejde-67 211 26 a a DET ejde-67 211 27 subsequence subsequence NOUN ejde-67 211 28 ( ( PUNCT ejde-67 211 29 sn)n sn)n PROPN ejde-67 211 30 of of ADP ejde-67 211 31 ( ( PUNCT ejde-67 211 32 s′n)n s′n)n NUM ejde-67 211 33 and and CCONJ ejde-67 211 34 a a DET ejde-67 211 35 function function NOUN ejde-67 211 36 g g ADP ejde-67 211 37 such such ADJ ejde-67 211 38 that that PRON ejde-67 211 39 , , PUNCT ejde-67 211 40 f(t+ f(t+ ADJ ejde-67 211 41 sn)→ sn)→ PROPN ejde-67 211 42 g(t g(t PROPN ejde-67 211 43 ) ) PUNCT ejde-67 211 44 as as SCONJ ejde-67 211 45 n→∞ n→∞ PRON ejde-67 211 46 uniformly uniformly ADV ejde-67 211 47 on on ADP ejde-67 211 48 r r NOUN ejde-67 211 49 we we PRON ejde-67 211 50 denote denote VERB ejde-67 211 51 by by ADP ejde-67 211 52 ap ap PROPN ejde-67 211 53 ( ( PUNCT ejde-67 211 54 r r NOUN ejde-67 211 55 , , PUNCT ejde-67 211 56 x x NOUN ejde-67 211 57 ) ) PUNCT ejde-67 211 58 the the DET ejde-67 211 59 set set NOUN ejde-67 211 60 of of ADP ejde-67 211 61 all all DET ejde-67 211 62 such such ADJ ejde-67 211 63 functions function NOUN ejde-67 211 64 . . PUNCT ejde-67 212 1 for for ADP ejde-67 212 2 some some DET ejde-67 212 3 preliminary preliminary ADJ ejde-67 212 4 results result NOUN ejde-67 212 5 on on ADP ejde-67 212 6 almost almost ADV ejde-67 212 7 periodic periodic ADJ ejde-67 212 8 functions function NOUN ejde-67 212 9 , , PUNCT ejde-67 212 10 we we PRON ejde-67 212 11 refer refer VERB ejde-67 212 12 the the DET ejde-67 212 13 reader reader NOUN ejde-67 212 14 to to ADP ejde-67 212 15 [ [ X ejde-67 212 16 20 20 NUM ejde-67 212 17 ] ] PUNCT ejde-67 212 18 . . PUNCT ejde-67 213 1 definition definition NOUN ejde-67 213 2 4.3 4.3 NUM ejde-67 213 3 ( ( PUNCT ejde-67 213 4 [ [ X ejde-67 213 5 17 17 NUM ejde-67 213 6 ] ] NUM ejde-67 213 7 ) ) PUNCT ejde-67 213 8 . . PUNCT ejde-67 214 1 the the DET ejde-67 214 2 bochner bochner NOUN ejde-67 214 3 transform transform VERB ejde-67 214 4 f f PROPN ejde-67 214 5 b b PROPN ejde-67 214 6 of of ADP ejde-67 214 7 a a DET ejde-67 214 8 function function NOUN ejde-67 214 9 f f PROPN ejde-67 214 10 ∈ ∈ PROPN ejde-67 214 11 lploc(r lploc(r PROPN ejde-67 214 12 , , PUNCT ejde-67 214 13 x x X ejde-67 214 14 ) ) PUNCT ejde-67 214 15 is be AUX ejde-67 214 16 the the DET ejde-67 214 17 function function NOUN ejde-67 214 18 f f PROPN ejde-67 214 19 b b PROPN ejde-67 214 20 : : PUNCT ejde-67 214 21 r→ r→ PROPN ejde-67 214 22 lploc([0 lploc([0 PROPN ejde-67 214 23 , , PUNCT ejde-67 214 24 1 1 NUM ejde-67 214 25 ] ] PUNCT ejde-67 214 26 , , PUNCT ejde-67 214 27 x x NOUN ejde-67 214 28 ) ) PUNCT ejde-67 214 29 , , PUNCT ejde-67 214 30 defined define VERB ejde-67 214 31 for for ADP ejde-67 214 32 each each DET ejde-67 214 33 t t NOUN ejde-67 214 34 ∈ ∈ NOUN ejde-67 214 35 r r NOUN ejde-67 214 36 by by ADP ejde-67 214 37 ( ( PUNCT ejde-67 214 38 f f PROPN ejde-67 214 39 b(t))(s b(t))(s PROPN ejde-67 214 40 ) ) PUNCT ejde-67 214 41 = = PUNCT ejde-67 215 1 f(t+ f(t+ NUM ejde-67 215 2 s s X ejde-67 215 3 ) ) PUNCT ejde-67 215 4 for for ADP ejde-67 215 5 s s X ejde-67 215 6 ∈ ∈ PROPN ejde-67 216 1 [ [ X ejde-67 216 2 0 0 NUM ejde-67 216 3 , , PUNCT ejde-67 216 4 1 1 NUM ejde-67 216 5 ] ] PUNCT ejde-67 216 6 . . PUNCT ejde-67 217 1 definition definition NOUN ejde-67 217 2 4.4 4.4 NUM ejde-67 217 3 ( ( PUNCT ejde-67 217 4 [ [ X ejde-67 217 5 17 17 NUM ejde-67 217 6 ] ] PUNCT ejde-67 217 7 ) ) PUNCT ejde-67 217 8 . . PUNCT ejde-67 218 1 let let VERB ejde-67 218 2 p p PRON ejde-67 218 3 ≥ ≥ NOUN ejde-67 218 4 1 1 NUM ejde-67 218 5 . . PUNCT ejde-67 219 1 the the DET ejde-67 219 2 space space NOUN ejde-67 219 3 bsp(r bsp(r PROPN ejde-67 219 4 , , PUNCT ejde-67 219 5 x x NOUN ejde-67 219 6 ) ) PUNCT ejde-67 219 7 consists consist VERB ejde-67 219 8 of of ADP ejde-67 219 9 all all DET ejde-67 219 10 functions function NOUN ejde-67 219 11 f f PROPN ejde-67 219 12 ∈ ∈ PROPN ejde-67 219 13 lploc(r lploc(r PROPN ejde-67 219 14 , , PUNCT ejde-67 219 15 x x NOUN ejde-67 219 16 ) ) PUNCT ejde-67 219 17 such such ADJ ejde-67 219 18 that that SCONJ ejde-67 219 19 f f PROPN ejde-67 219 20 b b PROPN ejde-67 219 21 : : PUNCT ejde-67 219 22 r→ r→ PROPN ejde-67 219 23 lploc([0 lploc([0 PROPN ejde-67 219 24 , , PUNCT ejde-67 219 25 1 1 NUM ejde-67 219 26 ] ] PUNCT ejde-67 219 27 , , PUNCT ejde-67 219 28 x x NOUN ejde-67 219 29 ) ) PUNCT ejde-67 219 30 , , PUNCT ejde-67 219 31 is be AUX ejde-67 219 32 bounded bound VERB ejde-67 219 33 ; ; PUNCT ejde-67 219 34 that that ADV ejde-67 219 35 is is ADV ejde-67 219 36 , , PUNCT ejde-67 219 37 sup sup NOUN ejde-67 219 38 t∈r t∈r NOUN ejde-67 219 39 ( ( PUNCT ejde-67 219 40 ∫ ∫ PROPN ejde-67 219 41 t+1 t+1 PROPN ejde-67 219 42 t t PROPN ejde-67 219 43 ‖f(s)‖pds ‖f(s)‖pds PROPN ejde-67 219 44 ) ) PUNCT ejde-67 219 45 1 1 X ejde-67 219 46 / / SYM ejde-67 219 47 p p X ejde-67 219 48 < < X ejde-67 219 49 ∞. ∞. PROPN ejde-67 219 50 this this PRON ejde-67 219 51 is be AUX ejde-67 219 52 a a DET ejde-67 219 53 normed normed ADJ ejde-67 219 54 space space NOUN ejde-67 219 55 when when SCONJ ejde-67 219 56 equipped equip VERB ejde-67 219 57 with with ADP ejde-67 219 58 the the DET ejde-67 219 59 norm norm NOUN ejde-67 219 60 ‖f‖bsp ‖f‖bsp NOUN ejde-67 220 1 = = PUNCT ejde-67 220 2 sup sup NOUN ejde-67 220 3 t∈r t∈r NOUN ejde-67 220 4 ( ( PUNCT ejde-67 220 5 ∫ ∫ PROPN ejde-67 220 6 t+1 t+1 PROPN ejde-67 220 7 t t PROPN ejde-67 220 8 ‖f(s)‖pds ‖f(s)‖pds PROPN ejde-67 220 9 ) ) PUNCT ejde-67 220 10 1 1 X ejde-67 220 11 / / SYM ejde-67 220 12 p p NOUN ejde-67 220 13 note note NOUN ejde-67 220 14 that that SCONJ ejde-67 220 15 the the DET ejde-67 220 16 functions function NOUN ejde-67 220 17 of of ADP ejde-67 220 18 bsp(r bsp(r PROPN ejde-67 220 19 , , PUNCT ejde-67 220 20 x x PRON ejde-67 220 21 ) ) PUNCT ejde-67 220 22 may may AUX ejde-67 220 23 not not PART ejde-67 220 24 be be AUX ejde-67 220 25 bounded bound VERB ejde-67 220 26 . . PUNCT ejde-67 221 1 ejde-2023/39 ejde-2023/39 ADJ ejde-67 221 2 reduction reduction NOUN ejde-67 221 3 principle principle NOUN ejde-67 221 4 9 9 NUM ejde-67 221 5 definition definition NOUN ejde-67 221 6 4.5 4.5 NUM ejde-67 221 7 ( ( PUNCT ejde-67 221 8 [ [ X ejde-67 221 9 8 8 NUM ejde-67 221 10 ] ] NUM ejde-67 221 11 ) ) PUNCT ejde-67 221 12 . . PUNCT ejde-67 222 1 a a DET ejde-67 222 2 function function NOUN ejde-67 222 3 f f PROPN ejde-67 222 4 ∈ ∈ PROPN ejde-67 222 5 lploc(r lploc(r PROPN ejde-67 222 6 , , PUNCT ejde-67 222 7 x x X ejde-67 222 8 ) ) PUNCT ejde-67 222 9 is be AUX ejde-67 222 10 sp sp NOUN ejde-67 222 11 - - PUNCT ejde-67 222 12 almost almost ADV ejde-67 222 13 periodic periodic ADJ ejde-67 222 14 if if SCONJ ejde-67 222 15 for for ADP ejde-67 222 16 every every DET ejde-67 222 17 sequence sequence NOUN ejde-67 222 18 of of ADP ejde-67 222 19 real real ADJ ejde-67 222 20 numbers number NOUN ejde-67 222 21 ( ( PUNCT ejde-67 222 22 s′n)n s′n)n NUM ejde-67 222 23 , , PUNCT ejde-67 222 24 there there PRON ejde-67 222 25 exist exist VERB ejde-67 222 26 a a DET ejde-67 222 27 subsequence subsequence NOUN ejde-67 222 28 ( ( PUNCT ejde-67 222 29 sn)n sn)n PROPN ejde-67 222 30 of of ADP ejde-67 222 31 ( ( PUNCT ejde-67 222 32 s′n)n s′n)n NUM ejde-67 222 33 and and CCONJ ejde-67 222 34 a a DET ejde-67 222 35 function function NOUN ejde-67 222 36 g g PROPN ejde-67 222 37 ∈ ∈ PROPN ejde-67 222 38 lploc(r lploc(r NOUN ejde-67 222 39 , , PUNCT ejde-67 222 40 x x NOUN ejde-67 222 41 ) ) PUNCT ejde-67 222 42 such such ADJ ejde-67 222 43 that that SCONJ ejde-67 222 44 , , PUNCT ejde-67 222 45 for for ADP ejde-67 222 46 each each DET ejde-67 222 47 t t NOUN ejde-67 222 48 ∈ ∈ PROPN ejde-67 222 49 r r NOUN ejde-67 222 50 , , PUNCT ejde-67 222 51 sup sup NOUN ejde-67 222 52 t∈r t∈r NOUN ejde-67 222 53 ( ( PUNCT ejde-67 222 54 ∫ ∫ PROPN ejde-67 222 55 t+1 t+1 PROPN ejde-67 222 56 t t PROPN ejde-67 222 57 ‖f(s+ ‖f(s+ PROPN ejde-67 222 58 sn)− sn)− VERB ejde-67 222 59 g(s)‖pds g(s)‖pds X ejde-67 222 60 ) ) PUNCT ejde-67 223 1 1 1 X ejde-67 223 2 / / SYM ejde-67 223 3 p p X ejde-67 223 4 → → SYM ejde-67 223 5 0 0 NUM ejde-67 223 6 as as SCONJ ejde-67 223 7 n→∞ n→∞ PRON ejde-67 223 8 let let VERB ejde-67 223 9 sap sap PROPN ejde-67 223 10 p(r p(r PROPN ejde-67 223 11 , , PUNCT ejde-67 223 12 x x X ejde-67 223 13 ) ) PUNCT ejde-67 223 14 denote denote VERB ejde-67 223 15 this this DET ejde-67 223 16 class class NOUN ejde-67 223 17 of of ADP ejde-67 223 18 functions function NOUN ejde-67 223 19 . . PUNCT ejde-67 224 1 using use VERB ejde-67 224 2 the the DET ejde-67 224 3 bochner bochner NOUN ejde-67 224 4 characterization characterization NOUN ejde-67 224 5 in in ADP ejde-67 224 6 theorem theorem ADJ ejde-67 224 7 4.2 4.2 NUM ejde-67 224 8 and and CCONJ ejde-67 224 9 the the DET ejde-67 224 10 completeness completeness NOUN ejde-67 224 11 of of ADP ejde-67 224 12 the the DET ejde-67 224 13 space space NOUN ejde-67 224 14 bsp(r bsp(r PROPN ejde-67 224 15 , , PUNCT ejde-67 224 16 x x NOUN ejde-67 224 17 ) ) PUNCT ejde-67 224 18 , , PUNCT ejde-67 224 19 we we PRON ejde-67 224 20 can can AUX ejde-67 224 21 see see VERB ejde-67 224 22 that that SCONJ ejde-67 224 23 f f PROPN ejde-67 224 24 ∈ ∈ PROPN ejde-67 224 25 ap ap PROPN ejde-67 224 26 p(r p(r PROPN ejde-67 224 27 , , PUNCT ejde-67 224 28 x x NOUN ejde-67 224 29 ) ) PUNCT ejde-67 225 1 if if SCONJ ejde-67 225 2 and and CCONJ ejde-67 225 3 only only ADV ejde-67 225 4 if if SCONJ ejde-67 225 5 f f PROPN ejde-67 225 6 b b PROPN ejde-67 225 7 ∈ ∈ PROPN ejde-67 225 8 ap ap PROPN ejde-67 225 9 p(r p(r PROPN ejde-67 225 10 , , PUNCT ejde-67 225 11 lp([0 lp([0 NOUN ejde-67 225 12 , , PUNCT ejde-67 225 13 1]x 1]x NUM ejde-67 225 14 ) ) PUNCT ejde-67 225 15 ) ) PUNCT ejde-67 225 16 . . PUNCT ejde-67 226 1 moreover moreover ADV ejde-67 226 2 , , PUNCT ejde-67 226 3 for for ADP ejde-67 226 4 all all DET ejde-67 226 5 p p PRON ejde-67 226 6 ≥ ≥ NUM ejde-67 226 7 1 1 NUM ejde-67 226 8 , , PUNCT ejde-67 226 9 ap ap AUX ejde-67 226 10 ( ( PUNCT ejde-67 226 11 r r NOUN ejde-67 226 12 , , PUNCT ejde-67 226 13 x x X ejde-67 226 14 ) ) PUNCT ejde-67 226 15 is be AUX ejde-67 226 16 a a DET ejde-67 226 17 subset subset NOUN ejde-67 226 18 of⊂ of⊂ PROPN ejde-67 226 19 sap sap PROPN ejde-67 226 20 p(r p(r PROPN ejde-67 226 21 , , PUNCT ejde-67 226 22 x x NOUN ejde-67 226 23 ) ) PUNCT ejde-67 226 24 . . PUNCT ejde-67 227 1 if if SCONJ ejde-67 227 2 p p PRON ejde-67 227 3 ≥ ≥ AUX ejde-67 227 4 q q NOUN ejde-67 227 5 , , PUNCT ejde-67 227 6 then then ADV ejde-67 227 7 sap sap PROPN ejde-67 227 8 p(r p(r PROPN ejde-67 227 9 , , PUNCT ejde-67 227 10 x x X ejde-67 227 11 ) ) PUNCT ejde-67 227 12 ⊂ ⊂ PROPN ejde-67 227 13 sap sap PROPN ejde-67 227 14 q(r q(r PROPN ejde-67 227 15 , , PUNCT ejde-67 227 16 x x NOUN ejde-67 227 17 ) ) PUNCT ejde-67 227 18 . . PUNCT ejde-67 228 1 definition definition NOUN ejde-67 228 2 4.6 4.6 NUM ejde-67 228 3 ( ( PUNCT ejde-67 228 4 [ [ X ejde-67 228 5 6 6 NUM ejde-67 228 6 ] ] NUM ejde-67 228 7 ) ) PUNCT ejde-67 228 8 . . PUNCT ejde-67 229 1 a a DET ejde-67 229 2 continuous continuous ADJ ejde-67 229 3 function function NOUN ejde-67 229 4 x x NOUN ejde-67 229 5 : : PUNCT ejde-67 229 6 r r X ejde-67 229 7 → → PUNCT ejde-67 229 8 x x X ejde-67 229 9 is be AUX ejde-67 229 10 said say VERB ejde-67 229 11 to to PART ejde-67 229 12 be be AUX ejde-67 229 13 almost almost ADV ejde-67 229 14 automorphic automorphic ADJ ejde-67 229 15 if if SCONJ ejde-67 229 16 for for ADP ejde-67 229 17 any any DET ejde-67 229 18 sequence sequence NOUN ejde-67 229 19 of of ADP ejde-67 229 20 real real ADJ ejde-67 229 21 numbers number NOUN ejde-67 229 22 ( ( PUNCT ejde-67 229 23 t′n)n t′n)n NUM ejde-67 229 24 , , PUNCT ejde-67 229 25 there there PRON ejde-67 229 26 exists exist VERB ejde-67 229 27 a a DET ejde-67 229 28 subsequence subsequence NOUN ejde-67 229 29 ( ( PUNCT ejde-67 229 30 tn)n tn)n PROPN ejde-67 229 31 of of ADP ejde-67 229 32 ( ( PUNCT ejde-67 229 33 t′n)n t′n)n X ejde-67 229 34 such such ADJ ejde-67 229 35 that that PRON ejde-67 229 36 y(t y(t NOUN ejde-67 229 37 ) ) PUNCT ejde-67 230 1 = = VERB ejde-67 230 2 lim lim PROPN ejde-67 230 3 n→+∞ n→+∞ PROPN ejde-67 230 4 x(t+ x(t+ PROPN ejde-67 230 5 tn tn PROPN ejde-67 230 6 ) ) PUNCT ejde-67 230 7 , , PUNCT ejde-67 230 8 ( ( PUNCT ejde-67 230 9 4.1 4.1 NUM ejde-67 230 10 ) ) PUNCT ejde-67 230 11 is be AUX ejde-67 230 12 well well ADV ejde-67 230 13 defined define VERB ejde-67 230 14 for for ADP ejde-67 230 15 each each DET ejde-67 230 16 t t NOUN ejde-67 230 17 ∈ ∈ PROPN ejde-67 230 18 r r NOUN ejde-67 230 19 and and CCONJ ejde-67 230 20 lim lim PROPN ejde-67 231 1 n→+∞ n→+∞ VERB ejde-67 231 2 y(t− y(t− PROPN ejde-67 231 3 tn tn PROPN ejde-67 231 4 ) ) PUNCT ejde-67 231 5 = = SYM ejde-67 232 1 x(t x(t PROPN ejde-67 232 2 ) ) PUNCT ejde-67 232 3 for for ADP ejde-67 232 4 all all DET ejde-67 232 5 t t PROPN ejde-67 232 6 ∈ ∈ PROPN ejde-67 232 7 r. r. PROPN ejde-67 232 8 ( ( PUNCT ejde-67 232 9 4.2 4.2 NUM ejde-67 232 10 ) ) PUNCT ejde-67 232 11 we we PRON ejde-67 232 12 denote denote VERB ejde-67 232 13 by by ADP ejde-67 232 14 aa(r aa(r PROPN ejde-67 232 15 , , PUNCT ejde-67 232 16 x x X ejde-67 232 17 ) ) PUNCT ejde-67 232 18 the the DET ejde-67 232 19 space space NOUN ejde-67 232 20 of of ADP ejde-67 232 21 all all DET ejde-67 232 22 almost almost ADV ejde-67 232 23 automorphic automorphic ADJ ejde-67 232 24 x x ADJ ejde-67 232 25 - - PUNCT ejde-67 232 26 valued value VERB ejde-67 232 27 functions function NOUN ejde-67 232 28 . . PUNCT ejde-67 233 1 moreover moreover ADV ejde-67 233 2 , , PUNCT ejde-67 233 3 if if SCONJ ejde-67 233 4 the the DET ejde-67 233 5 limits limit NOUN ejde-67 233 6 in in ADP ejde-67 233 7 ( ( PUNCT ejde-67 233 8 4.1 4.1 NUM ejde-67 233 9 ) ) PUNCT ejde-67 233 10 and and CCONJ ejde-67 233 11 ( ( PUNCT ejde-67 233 12 4.2 4.2 NUM ejde-67 233 13 ) ) PUNCT ejde-67 233 14 are be AUX ejde-67 233 15 uniform uniform ADJ ejde-67 233 16 on on ADP ejde-67 233 17 any any DET ejde-67 233 18 compact compact ADJ ejde-67 233 19 subset subset NOUN ejde-67 234 1 k k PROPN ejde-67 234 2 ⊂ ⊂ PROPN ejde-67 234 3 r r X ejde-67 234 4 , , PUNCT ejde-67 234 5 we we PRON ejde-67 234 6 say say VERB ejde-67 234 7 that that SCONJ ejde-67 234 8 x x PRON ejde-67 234 9 is be AUX ejde-67 234 10 compact compact ADJ ejde-67 234 11 almost almost ADV ejde-67 234 12 automorphic automorphic ADJ ejde-67 234 13 . . PUNCT ejde-67 235 1 if if SCONJ ejde-67 235 2 we we PRON ejde-67 235 3 denote denote VERB ejde-67 235 4 aac(r aac(r PROPN ejde-67 235 5 , , PUNCT ejde-67 235 6 x x X ejde-67 235 7 ) ) PUNCT ejde-67 235 8 the the DET ejde-67 235 9 space space NOUN ejde-67 235 10 of of ADP ejde-67 235 11 all all DET ejde-67 235 12 compact compact ADJ ejde-67 235 13 almost almost ADV ejde-67 235 14 automorphic automorphic ADJ ejde-67 235 15 x x ADJ ejde-67 235 16 - - PUNCT ejde-67 235 17 valued value VERB ejde-67 235 18 functions function NOUN ejde-67 235 19 , , PUNCT ejde-67 235 20 then then ADV ejde-67 235 21 we we PRON ejde-67 235 22 have have VERB ejde-67 235 23 ap ap PROPN ejde-67 235 24 ( ( PUNCT ejde-67 235 25 r r NOUN ejde-67 235 26 , , PUNCT ejde-67 235 27 x x NOUN ejde-67 235 28 ) ) PUNCT ejde-67 236 1 ⊂ ⊂ PROPN ejde-67 237 1 aac(r aac(r PROPN ejde-67 237 2 , , PUNCT ejde-67 237 3 x x X ejde-67 237 4 ) ) PUNCT ejde-67 237 5 ⊂ ⊂ PROPN ejde-67 237 6 aa(r aa(r PROPN ejde-67 237 7 , , PUNCT ejde-67 237 8 x x PRON ejde-67 237 9 ) ) PUNCT ejde-67 237 10 ⊂ ⊂ PROPN ejde-67 237 11 bc(r bc(r PROPN ejde-67 237 12 , , PUNCT ejde-67 237 13 x x NOUN ejde-67 237 14 ) ) PUNCT ejde-67 237 15 . . PUNCT ejde-67 238 1 example example NOUN ejde-67 239 1 4.7 4.7 NUM ejde-67 239 2 ( ( PUNCT ejde-67 239 3 [ [ X ejde-67 239 4 15 15 NUM ejde-67 239 5 ] ] NUM ejde-67 239 6 ) ) PUNCT ejde-67 239 7 . . PUNCT ejde-67 240 1 h(t h(t PROPN ejde-67 240 2 ) ) PUNCT ejde-67 241 1 = = PRON ejde-67 241 2 sin sin NOUN ejde-67 241 3 ( ( PUNCT ejde-67 241 4 1 1 NUM ejde-67 241 5 2+cos 2+co NOUN ejde-67 241 6 t+cos t+cos PART ejde-67 241 7 √ √ NUM ejde-67 241 8 2 2 NUM ejde-67 241 9 t t NOUN ejde-67 241 10 ) ) PUNCT ejde-67 241 11 is be AUX ejde-67 241 12 an an DET ejde-67 241 13 almost almost ADV ejde-67 241 14 automorphic automorphic ADJ ejde-67 241 15 function function NOUN ejde-67 241 16 but but CCONJ ejde-67 241 17 it it PRON ejde-67 241 18 is be AUX ejde-67 241 19 not not PART ejde-67 241 20 almost almost ADV ejde-67 241 21 periodic periodic ADJ ejde-67 241 22 . . PUNCT ejde-67 242 1 since since SCONJ ejde-67 242 2 it it PRON ejde-67 242 3 is be AUX ejde-67 242 4 not not PART ejde-67 242 5 uniformly uniformly ADV ejde-67 242 6 continuous continuous ADJ ejde-67 242 7 . . PUNCT ejde-67 243 1 definition definition NOUN ejde-67 243 2 4.8 4.8 NUM ejde-67 243 3 ( ( PUNCT ejde-67 243 4 [ [ X ejde-67 243 5 9 9 NUM ejde-67 243 6 ] ] NUM ejde-67 243 7 ) ) PUNCT ejde-67 243 8 . . PUNCT ejde-67 244 1 a a DET ejde-67 244 2 function function NOUN ejde-67 244 3 f f PROPN ejde-67 244 4 ∈ ∈ PROPN ejde-67 244 5 lploc(r lploc(r PROPN ejde-67 244 6 , , PUNCT ejde-67 244 7 x x PRON ejde-67 244 8 ) ) PUNCT ejde-67 244 9 is be AUX ejde-67 244 10 said say VERB ejde-67 244 11 to to PART ejde-67 244 12 be be AUX ejde-67 244 13 sp sp VERB ejde-67 244 14 - - PUNCT ejde-67 244 15 almost almost ADV ejde-67 244 16 automorphic automorphic ADJ ejde-67 244 17 for for ADP ejde-67 244 18 some some DET ejde-67 244 19 p p NOUN ejde-67 244 20 ≥ ≥ NUM ejde-67 244 21 1 1 NUM ejde-67 244 22 if if SCONJ ejde-67 244 23 the the DET ejde-67 244 24 function function NOUN ejde-67 244 25 f f PROPN ejde-67 244 26 b b PROPN ejde-67 244 27 : : PUNCT ejde-67 244 28 r→ r→ PROPN ejde-67 244 29 lploc([0 lploc([0 PROPN ejde-67 244 30 , , PUNCT ejde-67 244 31 1 1 NUM ejde-67 244 32 ] ] PUNCT ejde-67 244 33 , , PUNCT ejde-67 244 34 x x X ejde-67 244 35 ) ) PUNCT ejde-67 244 36 is be AUX ejde-67 244 37 almost almost ADV ejde-67 244 38 automorphic automorphic ADJ ejde-67 244 39 . . PUNCT ejde-67 245 1 the the DET ejde-67 245 2 following follow VERB ejde-67 245 3 characterization characterization NOUN ejde-67 245 4 of of ADP ejde-67 245 5 almost almost ADV ejde-67 245 6 automorphy automorphy NOUN ejde-67 245 7 in in ADP ejde-67 245 8 the the DET ejde-67 245 9 sense sense NOUN ejde-67 245 10 of of ADP ejde-67 245 11 stepanov stepanov NOUN ejde-67 245 12 is be AUX ejde-67 245 13 essential essential ADJ ejde-67 245 14 for for ADP ejde-67 245 15 the the DET ejde-67 245 16 remainder remainder NOUN ejde-67 245 17 of of ADP ejde-67 245 18 this this DET ejde-67 245 19 work work NOUN ejde-67 245 20 . . PUNCT ejde-67 246 1 proposition proposition NOUN ejde-67 246 2 4.9 4.9 NUM ejde-67 246 3 ( ( PUNCT ejde-67 246 4 [ [ X ejde-67 246 5 17 17 NUM ejde-67 246 6 ] ] NUM ejde-67 246 7 ) ) PUNCT ejde-67 246 8 . . PUNCT ejde-67 247 1 a a DET ejde-67 247 2 function function NOUN ejde-67 247 3 f f PROPN ejde-67 247 4 ∈ ∈ PROPN ejde-67 247 5 lploc(r lploc(r PROPN ejde-67 247 6 , , PUNCT ejde-67 247 7 x x X ejde-67 247 8 ) ) PUNCT ejde-67 247 9 is be AUX ejde-67 247 10 sp sp NOUN ejde-67 247 11 - - PUNCT ejde-67 247 12 almost almost ADV ejde-67 247 13 automorphic automorphic ADJ ejde-67 247 14 if if SCONJ ejde-67 247 15 and and CCONJ ejde-67 247 16 only only ADV ejde-67 247 17 if if SCONJ ejde-67 247 18 , , PUNCT ejde-67 247 19 for for ADP ejde-67 247 20 every every DET ejde-67 247 21 sequence sequence NOUN ejde-67 247 22 of of ADP ejde-67 247 23 real real ADJ ejde-67 247 24 numbers number NOUN ejde-67 247 25 ( ( PUNCT ejde-67 247 26 s′n)n s′n)n NUM ejde-67 247 27 , , PUNCT ejde-67 247 28 there there PRON ejde-67 247 29 exist exist VERB ejde-67 247 30 a a DET ejde-67 247 31 subsequence subsequence NOUN ejde-67 247 32 ( ( PUNCT ejde-67 247 33 sn)n sn)n PROPN ejde-67 247 34 of of ADP ejde-67 247 35 ( ( PUNCT ejde-67 247 36 s′n)n s′n)n NUM ejde-67 247 37 and and CCONJ ejde-67 247 38 a a DET ejde-67 247 39 function function NOUN ejde-67 247 40 g g PROPN ejde-67 247 41 ∈ ∈ PROPN ejde-67 247 42 lploc(r lploc(r NOUN ejde-67 247 43 , , PUNCT ejde-67 247 44 x x NOUN ejde-67 247 45 ) ) PUNCT ejde-67 247 46 such such ADJ ejde-67 247 47 that that SCONJ ejde-67 247 48 , , PUNCT ejde-67 247 49 for for ADP ejde-67 247 50 each each DET ejde-67 247 51 t t PROPN ejde-67 247 52 ∈ ∈ PROPN ejde-67 247 53 r,(∫ r,(∫ NOUN ejde-67 247 54 t+1 t+1 PROPN ejde-67 247 55 t t PROPN ejde-67 247 56 ‖f(s+ ‖f(s+ PROPN ejde-67 247 57 sn)− sn)− VERB ejde-67 247 58 g(s)‖pds g(s)‖pds X ejde-67 247 59 ) ) PUNCT ejde-67 248 1 1 1 X ejde-67 248 2 / / SYM ejde-67 248 3 p p X ejde-67 248 4 → → SYM ejde-67 248 5 0 0 PUNCT ejde-67 248 6 as as ADP ejde-67 248 7 n→∞,(∫ n→∞,(∫ PROPN ejde-67 248 8 t+1 t+1 PROPN ejde-67 248 9 t t PROPN ejde-67 248 10 ‖g(s− ‖g(s− PROPN ejde-67 248 11 sn)− sn)− NOUN ejde-67 248 12 f(s)‖pds f(s)‖pds NUM ejde-67 248 13 ) ) PUNCT ejde-67 248 14 1 1 NUM ejde-67 248 15 / / SYM ejde-67 248 16 p p X ejde-67 248 17 → → SYM ejde-67 248 18 0 0 NUM ejde-67 248 19 as as SCONJ ejde-67 248 20 n→∞ n→∞ PRON ejde-67 248 21 let let VERB ejde-67 248 22 saap(r saap(r PROPN ejde-67 248 23 , , PUNCT ejde-67 248 24 x x NOUN ejde-67 248 25 ) ) PUNCT ejde-67 248 26 denote denote VERB ejde-67 248 27 the the DET ejde-67 248 28 space space NOUN ejde-67 248 29 of of ADP ejde-67 248 30 sp sp NOUN ejde-67 248 31 - - PUNCT ejde-67 248 32 almost almost ADV ejde-67 248 33 automorphic automorphic ADJ ejde-67 248 34 x x ADJ ejde-67 248 35 - - PUNCT ejde-67 248 36 valued value VERB ejde-67 248 37 functions function NOUN ejde-67 248 38 on on ADP ejde-67 248 39 r. r. PROPN ejde-67 248 40 then then ADV ejde-67 248 41 , , PUNCT ejde-67 248 42 for for ADP ejde-67 248 43 all all DET ejde-67 248 44 p p PRON ejde-67 248 45 ≥ ≥ NOUN ejde-67 248 46 1 1 NUM ejde-67 248 47 , , PUNCT ejde-67 248 48 we we PRON ejde-67 248 49 have have VERB ejde-67 248 50 aa(r aa(r NOUN ejde-67 248 51 , , PUNCT ejde-67 248 52 x x PRON ejde-67 248 53 ) ) PUNCT ejde-67 248 54 ⊂ ⊂ PROPN ejde-67 248 55 saap(r saap(r PROPN ejde-67 248 56 , , PUNCT ejde-67 248 57 x x NOUN ejde-67 248 58 ) ) PUNCT ejde-67 248 59 . . PUNCT ejde-67 249 1 moreover moreover ADV ejde-67 249 2 , , PUNCT ejde-67 249 3 if if SCONJ ejde-67 249 4 p p PRON ejde-67 249 5 ≥ ≥ NOUN ejde-67 249 6 q q NOUN ejde-67 249 7 , , PUNCT ejde-67 249 8 then then ADV ejde-67 249 9 saap(r saap(r PROPN ejde-67 249 10 , , PUNCT ejde-67 249 11 x x PRON ejde-67 249 12 ) ) PUNCT ejde-67 249 13 ⊂ ⊂ PROPN ejde-67 249 14 saaq(r saaq(r NOUN ejde-67 249 15 , , PUNCT ejde-67 249 16 x x NOUN ejde-67 249 17 ) ) PUNCT ejde-67 249 18 . . PUNCT ejde-67 250 1 if if SCONJ ejde-67 250 2 h h PROPN ejde-67 250 3 ∈ ∈ PROPN ejde-67 250 4 aa(r aa(r PROPN ejde-67 250 5 , , PUNCT ejde-67 250 6 c c NOUN ejde-67 250 7 ) ) PUNCT ejde-67 250 8 and and CCONJ ejde-67 250 9 f f PROPN ejde-67 250 10 ∈ ∈ PROPN ejde-67 250 11 saap(r saap(r PROPN ejde-67 250 12 , , PUNCT ejde-67 250 13 c c NOUN ejde-67 250 14 ) ) PUNCT ejde-67 250 15 , , PUNCT ejde-67 250 16 then then ADV ejde-67 250 17 hf hf PROPN ejde-67 250 18 ∈ ∈ PROPN ejde-67 250 19 saap(r saap(r PROPN ejde-67 250 20 , , PUNCT ejde-67 250 21 c c NOUN ejde-67 250 22 ) ) PUNCT ejde-67 250 23 . . PUNCT ejde-67 251 1 10 10 NUM ejde-67 251 2 m. m. NOUN ejde-67 251 3 el el PROPN ejde-67 251 4 attaouy attaouy PROPN ejde-67 251 5 , , PUNCT ejde-67 251 6 k. k. PROPN ejde-67 251 7 ezzinbi ezzinbi PROPN ejde-67 251 8 , , PUNCT ejde-67 251 9 g. g. PROPN ejde-67 251 10 m. m. PROPN ejde-67 251 11 n’guérékata n’guérékata PROPN ejde-67 251 12 ejde-2023/39 ejde-2023/39 VERB ejde-67 252 1 5 5 NUM ejde-67 252 2 . . PUNCT ejde-67 253 1 existence existence NOUN ejde-67 253 2 of of ADP ejde-67 253 3 almost almost ADV ejde-67 253 4 automorphic automorphic ADJ ejde-67 253 5 and and CCONJ ejde-67 253 6 almost almost ADV ejde-67 253 7 periodic periodic ADJ ejde-67 253 8 solutions solution NOUN ejde-67 253 9 the the DET ejde-67 253 10 aim aim NOUN ejde-67 253 11 of of ADP ejde-67 253 12 this this DET ejde-67 253 13 section section NOUN ejde-67 253 14 is be AUX ejde-67 253 15 to to PART ejde-67 253 16 study study VERB ejde-67 253 17 the the DET ejde-67 253 18 existence existence NOUN ejde-67 253 19 of of ADP ejde-67 253 20 an an DET ejde-67 253 21 almost almost ADV ejde-67 253 22 automorphic automorphic ADJ ejde-67 253 23 and and CCONJ ejde-67 253 24 almost almost ADV ejde-67 253 25 periodic periodic ADJ ejde-67 253 26 solution solution NOUN ejde-67 253 27 of of ADP ejde-67 253 28 ( ( PUNCT ejde-67 253 29 1.3 1.3 NUM ejde-67 253 30 ) ) PUNCT ejde-67 253 31 . . PUNCT ejde-67 254 1 in in ADP ejde-67 254 2 the the DET ejde-67 254 3 rest rest NOUN ejde-67 254 4 of of ADP ejde-67 254 5 this this DET ejde-67 254 6 section section NOUN ejde-67 254 7 , , PUNCT ejde-67 254 8 we we PRON ejde-67 254 9 assume assume VERB ejde-67 254 10 that that SCONJ ejde-67 254 11 a a PRON ejde-67 254 12 , , PUNCT ejde-67 254 13 l l NOUN ejde-67 254 14 and and CCONJ ejde-67 254 15 f f PROPN ejde-67 254 16 satisfy satisfy VERB ejde-67 254 17 the the DET ejde-67 254 18 conditions condition NOUN ejde-67 254 19 in in ADP ejde-67 254 20 section section NOUN ejde-67 254 21 2 2 NUM ejde-67 254 22 . . PUNCT ejde-67 255 1 we we PRON ejde-67 255 2 consider consider VERB ejde-67 255 3 the the DET ejde-67 255 4 ordinary ordinary ADJ ejde-67 255 5 differential differential ADJ ejde-67 255 6 equation equation NOUN ejde-67 255 7 z′(t z′(t NOUN ejde-67 255 8 ) ) PUNCT ejde-67 255 9 = = SYM ejde-67 255 10 bz(t bz(t X ejde-67 255 11 ) ) PUNCT ejde-67 255 12 + + NUM ejde-67 255 13 g(t g(t PROPN ejde-67 255 14 ) ) PUNCT ejde-67 255 15 , , PUNCT ejde-67 255 16 for for ADP ejde-67 255 17 t t PROPN ejde-67 255 18 ∈ ∈ PROPN ejde-67 255 19 r r PROPN ejde-67 255 20 , , PUNCT ejde-67 255 21 ( ( PUNCT ejde-67 255 22 5.1 5.1 NUM ejde-67 255 23 ) ) PUNCT ejde-67 255 24 where where SCONJ ejde-67 255 25 b b NOUN ejde-67 255 26 is be AUX ejde-67 255 27 a a DET ejde-67 255 28 matrix matrix NOUN ejde-67 255 29 and and CCONJ ejde-67 255 30 g g NOUN ejde-67 255 31 : : PUNCT ejde-67 255 32 r→ r→ PROPN ejde-67 255 33 rd rd PROPN ejde-67 255 34 . . PROPN ejde-67 255 35 theorem theorem VERB ejde-67 255 36 5.1 5.1 NUM ejde-67 255 37 ( ( PUNCT ejde-67 255 38 [ [ X ejde-67 255 39 1 1 NUM ejde-67 255 40 ] ] PUNCT ejde-67 255 41 ) ) PUNCT ejde-67 255 42 . . PUNCT ejde-67 256 1 assume assume VERB ejde-67 256 2 that that SCONJ ejde-67 256 3 g g PROPN ejde-67 256 4 is be AUX ejde-67 256 5 a a DET ejde-67 256 6 s1 s1 NOUN ejde-67 256 7 - - PUNCT ejde-67 256 8 almost almost ADV ejde-67 256 9 periodic periodic ADJ ejde-67 256 10 function function NOUN ejde-67 256 11 . . PUNCT ejde-67 257 1 if if SCONJ ejde-67 257 2 ( ( PUNCT ejde-67 257 3 5.1 5.1 NUM ejde-67 257 4 ) ) PUNCT ejde-67 257 5 has have VERB ejde-67 257 6 a a DET ejde-67 257 7 bounded bound VERB ejde-67 257 8 solution solution NOUN ejde-67 257 9 on on ADP ejde-67 257 10 r+ r+ NOUN ejde-67 257 11 then then ADV ejde-67 257 12 it it PRON ejde-67 257 13 admits admit VERB ejde-67 257 14 an an DET ejde-67 257 15 almost almost ADV ejde-67 257 16 periodic periodic ADJ ejde-67 257 17 solution solution NOUN ejde-67 257 18 on on ADP ejde-67 257 19 r. r. PROPN ejde-67 257 20 now now ADV ejde-67 257 21 , , PUNCT ejde-67 257 22 we we PRON ejde-67 257 23 are be AUX ejde-67 257 24 able able ADJ ejde-67 257 25 to to PART ejde-67 257 26 establish establish VERB ejde-67 257 27 one one NUM ejde-67 257 28 of of ADP ejde-67 257 29 the the DET ejde-67 257 30 main main ADJ ejde-67 257 31 results result NOUN ejde-67 257 32 of of ADP ejde-67 257 33 this this DET ejde-67 257 34 work work NOUN ejde-67 257 35 . . PUNCT ejde-67 258 1 theorem theorem VERB ejde-67 258 2 5.2 5.2 NUM ejde-67 258 3 . . PUNCT ejde-67 259 1 assume assume VERB ejde-67 259 2 that that SCONJ ejde-67 259 3 the the DET ejde-67 259 4 semigroup semigroup NOUN ejde-67 259 5 ( ( PUNCT ejde-67 259 6 t t PROPN ejde-67 259 7 ( ( PUNCT ejde-67 259 8 t))t≥0 t))t≥0 PROPN ejde-67 259 9 is be AUX ejde-67 259 10 exponentially exponentially ADV ejde-67 259 11 stable stable ADJ ejde-67 259 12 and and CCONJ ejde-67 259 13 that that SCONJ ejde-67 259 14 the the DET ejde-67 259 15 operator operator NOUN ejde-67 259 16 t t PROPN ejde-67 259 17 ( ( PUNCT ejde-67 259 18 t)l t)l NOUN ejde-67 259 19 is be AUX ejde-67 259 20 compact compact ADJ ejde-67 259 21 for for ADP ejde-67 259 22 t t PROPN ejde-67 259 23 > > X ejde-67 259 24 0 0 PROPN ejde-67 259 25 . . PUNCT ejde-67 260 1 if if SCONJ ejde-67 260 2 f f PROPN ejde-67 260 3 : : PUNCT ejde-67 260 4 r→ r→ PROPN ejde-67 260 5 x x X ejde-67 260 6 is be AUX ejde-67 260 7 a a DET ejde-67 260 8 s1 s1 NOUN ejde-67 260 9 - - PUNCT ejde-67 260 10 almost almost ADV ejde-67 260 11 periodic periodic ADJ ejde-67 260 12 function function NOUN ejde-67 260 13 and and CCONJ ejde-67 260 14 ( ( PUNCT ejde-67 260 15 1.3 1.3 NUM ejde-67 260 16 ) ) PUNCT ejde-67 260 17 has have VERB ejde-67 260 18 a a DET ejde-67 260 19 bounded bound VERB ejde-67 260 20 solution solution NOUN ejde-67 260 21 on on ADP ejde-67 260 22 r+ r+ NOUN ejde-67 260 23 , , PUNCT ejde-67 260 24 then then ADV ejde-67 260 25 it it PRON ejde-67 260 26 has have VERB ejde-67 260 27 an an DET ejde-67 260 28 almost almost ADV ejde-67 260 29 periodic periodic ADJ ejde-67 260 30 solution solution NOUN ejde-67 260 31 . . PUNCT ejde-67 261 1 proof proof NOUN ejde-67 261 2 . . PUNCT ejde-67 262 1 let let VERB ejde-67 262 2 x x PRON ejde-67 262 3 be be AUX ejde-67 262 4 the the DET ejde-67 262 5 mild mild ADJ ejde-67 262 6 solution solution NOUN ejde-67 262 7 of of ADP ejde-67 262 8 ( ( PUNCT ejde-67 262 9 1.3 1.3 NUM ejde-67 262 10 ) ) PUNCT ejde-67 262 11 given give VERB ejde-67 262 12 by by ADP ejde-67 262 13 ( ( PUNCT ejde-67 262 14 3.4 3.4 NUM ejde-67 262 15 ) ) PUNCT ejde-67 262 16 . . PUNCT ejde-67 263 1 since since SCONJ ejde-67 263 2 z(t z(t NOUN ejde-67 263 3 ) ) PUNCT ejde-67 263 4 satisfies satisfie NOUN ejde-67 263 5 ( ( PUNCT ejde-67 263 6 3.3 3.3 NUM ejde-67 263 7 ) ) PUNCT ejde-67 263 8 , , PUNCT ejde-67 263 9 z z X ejde-67 263 10 ( ( PUNCT ejde-67 263 11 · · PUNCT ejde-67 263 12 ) ) PUNCT ejde-67 263 13 is be AUX ejde-67 263 14 bounded bound VERB ejde-67 263 15 on on ADP ejde-67 263 16 r+ r+ X ejde-67 263 17 . . PUNCT ejde-67 264 1 moreover moreover ADV ejde-67 264 2 , , PUNCT ejde-67 264 3 the the DET ejde-67 264 4 function function NOUN ejde-67 264 5 g(t g(t PROPN ejde-67 264 6 ) ) PUNCT ejde-67 265 1 = = PUNCT ejde-67 265 2 〈 〈 NOUN ejde-67 265 3 x∗ x∗ PROPN ejde-67 265 4 , , PUNCT ejde-67 265 5 f(t f(t PROPN ejde-67 265 6 ) ) PUNCT ejde-67 265 7 〉 〉 PROPN ejde-67 265 8 is be AUX ejde-67 265 9 s1 s1 NOUN ejde-67 265 10 - - PUNCT ejde-67 265 11 almost almost ADV ejde-67 265 12 periodic periodic NOUN ejde-67 265 13 . . PUNCT ejde-67 266 1 by by ADP ejde-67 266 2 theorem theorem NOUN ejde-67 266 3 5.1 5.1 NUM ejde-67 266 4 , , PUNCT ejde-67 266 5 we we PRON ejde-67 266 6 obtain obtain VERB ejde-67 266 7 that that SCONJ ejde-67 266 8 z z NOUN ejde-67 266 9 ( ( PUNCT ejde-67 266 10 · · PUNCT ejde-67 266 11 ) ) PUNCT ejde-67 266 12 is be AUX ejde-67 266 13 almost almost ADV ejde-67 266 14 periodic periodic ADJ ejde-67 266 15 on on ADP ejde-67 266 16 r r NOUN ejde-67 266 17 and and CCONJ ejde-67 266 18 φ φ PROPN ejde-67 266 19 z z PROPN ejde-67 266 20 ( ( PUNCT ejde-67 266 21 · · PUNCT ejde-67 266 22 ) ) PUNCT ejde-67 266 23 is be AUX ejde-67 266 24 an an DET ejde-67 266 25 almost almost ADV ejde-67 266 26 periodic periodic ADJ ejde-67 266 27 function function NOUN ejde-67 266 28 on on ADP ejde-67 266 29 r. r. PROPN ejde-67 266 30 let let VERB ejde-67 266 31 y y PROPN ejde-67 266 32 ( ( PUNCT ejde-67 266 33 t t PROPN ejde-67 266 34 ) ) PUNCT ejde-67 267 1 = = PROPN ejde-67 267 2 lim lim PROPN ejde-67 267 3 n→∞ n→∞ NUM ejde-67 268 1 ∫ ∫ PROPN ejde-67 268 2 t t PROPN ejde-67 268 3 −∞ −∞ ADP ejde-67 268 4 v v PROPN ejde-67 268 5 s(t− s(t− PROPN ejde-67 268 6 ξ)πs(b̃nx0f(ξ))dξ ξ)πs(b̃nx0f(ξ))dξ NUM ejde-67 268 7 , , PUNCT ejde-67 268 8 for for ADP ejde-67 268 9 t t PROPN ejde-67 268 10 ∈ ∈ PROPN ejde-67 268 11 r. r. PROPN ejde-67 268 12 ( ( PUNCT ejde-67 268 13 5.2 5.2 NUM ejde-67 268 14 ) ) PUNCT ejde-67 268 15 since since SCONJ ejde-67 268 16 f f PROPN ejde-67 268 17 is be AUX ejde-67 268 18 s1 s1 NOUN ejde-67 268 19 - - PUNCT ejde-67 268 20 almost almost ADV ejde-67 268 21 periodic periodic ADJ ejde-67 268 22 , , PUNCT ejde-67 268 23 for for ADP ejde-67 268 24 any any DET ejde-67 268 25 sequence sequence NOUN ejde-67 268 26 of of ADP ejde-67 268 27 real real ADJ ejde-67 268 28 numbers number NOUN ejde-67 268 29 ( ( PUNCT ejde-67 268 30 s′p)p s′p)p X ejde-67 268 31 there there PRON ejde-67 268 32 exists exist VERB ejde-67 268 33 a a DET ejde-67 268 34 subsequence subsequence NOUN ejde-67 268 35 ( ( PUNCT ejde-67 268 36 sp)p sp)p NOUN ejde-67 268 37 of of ADP ejde-67 268 38 ( ( PUNCT ejde-67 268 39 s′p)p s′p)p NOUN ejde-67 268 40 and and CCONJ ejde-67 268 41 a a DET ejde-67 268 42 function function NOUN ejde-67 268 43 g g PROPN ejde-67 268 44 ∈ ∈ PROPN ejde-67 268 45 lploc(r lploc(r NOUN ejde-67 268 46 , , PUNCT ejde-67 268 47 x x NOUN ejde-67 268 48 ) ) PUNCT ejde-67 268 49 such such ADJ ejde-67 268 50 that that SCONJ ejde-67 268 51 , , PUNCT ejde-67 268 52 for for ADP ejde-67 268 53 each each DET ejde-67 268 54 t t NOUN ejde-67 268 55 ∈ ∈ PROPN ejde-67 268 56 r r NOUN ejde-67 268 57 , , PUNCT ejde-67 268 58 sup sup NOUN ejde-67 268 59 t∈r t∈r NOUN ejde-67 268 60 ∫ ∫ NOUN ejde-67 268 61 t+1 t+1 PROPN ejde-67 268 62 t t PROPN ejde-67 268 63 ‖f(s+ ‖f(s+ ADJ ejde-67 268 64 sp)− sp)− PUNCT ejde-67 269 1 g(s)‖ds→ g(s)‖ds→ NOUN ejde-67 269 2 0 0 NUM ejde-67 269 3 as as ADP ejde-67 269 4 p→∞ p→∞ ADJ ejde-67 269 5 ( ( PUNCT ejde-67 269 6 5.3 5.3 NUM ejde-67 269 7 ) ) PUNCT ejde-67 269 8 on on ADP ejde-67 269 9 the the DET ejde-67 269 10 other other ADJ ejde-67 269 11 hand hand NOUN ejde-67 269 12 , , PUNCT ejde-67 269 13 let let VERB ejde-67 269 14 yk(t yk(t PRON ejde-67 269 15 ) ) PUNCT ejde-67 270 1 = = SYM ejde-67 270 2 lim lim PROPN ejde-67 270 3 n→∞ n→∞ NUM ejde-67 271 1 ∫ ∫ PROPN ejde-67 271 2 k k PROPN ejde-67 271 3 k−1 k−1 PROPN ejde-67 271 4 v v ADP ejde-67 271 5 s(ξ)πs(b̃nx0f(t− s(ξ)πs(b̃nx0f(t− NOUN ejde-67 271 6 ξ))dξ ξ))dξ PROPN ejde-67 271 7 , , PUNCT ejde-67 271 8 for for ADP ejde-67 271 9 t t PROPN ejde-67 271 10 ∈ ∈ PROPN ejde-67 271 11 r. r. PROPN ejde-67 271 12 first first ADV ejde-67 271 13 , , PUNCT ejde-67 271 14 we we PRON ejde-67 271 15 show show VERB ejde-67 271 16 that that SCONJ ejde-67 271 17 the the DET ejde-67 271 18 function function NOUN ejde-67 271 19 yk yk NOUN ejde-67 271 20 : : PUNCT ejde-67 271 21 t→ t→ NOUN ejde-67 271 22 yk(t yk(t NUM ejde-67 271 23 ) ) PUNCT ejde-67 271 24 is be AUX ejde-67 271 25 continuous continuous ADJ ejde-67 271 26 on on ADP ejde-67 271 27 r. r. PROPN ejde-67 271 28 let let VERB ejde-67 271 29 zk(t zk(t NOUN ejde-67 271 30 ) ) PUNCT ejde-67 271 31 = = SYM ejde-67 271 32 lim lim PROPN ejde-67 271 33 n→∞ n→∞ NUM ejde-67 272 1 ∫ ∫ PROPN ejde-67 272 2 k k PROPN ejde-67 272 3 k−1 k−1 PROPN ejde-67 272 4 v v PROPN ejde-67 272 5 s(ξ)πs(b̃nx0g(t− s(ξ)πs(b̃nx0g(t− NOUN ejde-67 272 6 ξ))dξ ξ))dξ PROPN ejde-67 272 7 , , PUNCT ejde-67 272 8 for for ADP ejde-67 272 9 t t PROPN ejde-67 272 10 ∈ ∈ PROPN ejde-67 272 11 r. r. PROPN ejde-67 272 12 we we PRON ejde-67 272 13 obtain obtain VERB ejde-67 272 14 that that DET ejde-67 272 15 ‖yk(t+ ‖yk(t+ ADJ ejde-67 272 16 sp)− sp)− ADJ ejde-67 273 1 zk(t)‖ zk(t)‖ NOUN ejde-67 273 2 = = SYM ejde-67 273 3 ‖ ‖ PROPN ejde-67 273 4 lim lim PROPN ejde-67 273 5 n→∞ n→∞ NUM ejde-67 274 1 ∫ ∫ PROPN ejde-67 274 2 k k PROPN ejde-67 274 3 k−1 k−1 PROPN ejde-67 274 4 v v ADP ejde-67 274 5 s(ξ)πs(b̃nx0(f(t+ s(ξ)πs(b̃nx0(f(t+ NOUN ejde-67 274 6 sp sp ADP ejde-67 274 7 − − PROPN ejde-67 274 8 ξ)− ξ)− PROPN ejde-67 274 9 g(t− g(t− PROPN ejde-67 274 10 ξ ξ PROPN ejde-67 274 11 ) ) PUNCT ejde-67 274 12 ) ) PUNCT ejde-67 274 13 ) ) PUNCT ejde-67 275 1 dξ‖ dξ‖ NOUN ejde-67 276 1 ≤mm̃‖πs‖ ≤mm̃‖πs‖ ADJ ejde-67 276 2 ∫ ∫ PROPN ejde-67 277 1 k k PROPN ejde-67 277 2 k−1 k−1 PROPN ejde-67 277 3 e−αξ‖f(t+ e−αξ‖f(t+ PROPN ejde-67 277 4 sp sp ADP ejde-67 277 5 − − PROPN ejde-67 277 6 ξ)− ξ)− PROPN ejde-67 277 7 g(t− g(t− PROPN ejde-67 277 8 ξ)‖ ξ)‖ NOUN ejde-67 277 9 dξ dξ PROPN ejde-67 277 10 ≤mm̃‖πs‖e−α(k−1 ≤mm̃‖πs‖e−α(k−1 PROPN ejde-67 277 11 ) ) PUNCT ejde-67 277 12 ∫ ∫ PROPN ejde-67 278 1 k k PROPN ejde-67 278 2 k−1 k−1 PROPN ejde-67 278 3 ‖f(t+ ‖f(t+ PROPN ejde-67 278 4 sp sp ADP ejde-67 278 5 − − PROPN ejde-67 278 6 ξ)− ξ)− PROPN ejde-67 278 7 g(t− g(t− PROPN ejde-67 278 8 ξ)‖ ξ)‖ VERB ejde-67 278 9 dξ dξ PROPN ejde-67 278 10 = = PROPN ejde-67 278 11 mm̃‖πs‖e−α(k−1 mm̃‖πs‖e−α(k−1 PROPN ejde-67 278 12 ) ) PUNCT ejde-67 278 13 ∫ ∫ PROPN ejde-67 279 1 t−k+1 t−k+1 VERB ejde-67 279 2 t−k t−k NOUN ejde-67 279 3 ‖f(sp ‖f(sp PROPN ejde-67 280 1 + + CCONJ ejde-67 280 2 ξ)− ξ)− PROPN ejde-67 280 3 g(ξ)‖ g(ξ)‖ PUNCT ejde-67 280 4 dξ dξ PROPN ejde-67 281 1 ≤mm̃‖πs‖e−α(k−1)sup ≤mm̃‖πs‖e−α(k−1)sup PROPN ejde-67 281 2 s∈r s∈r PROPN ejde-67 281 3 ∫ ∫ PROPN ejde-67 281 4 s s PART ejde-67 282 1 s+1 s+1 PROPN ejde-67 282 2 ‖f(sp ‖f(sp PROPN ejde-67 282 3 + + CCONJ ejde-67 282 4 ξ)− ξ)− PROPN ejde-67 282 5 g(ξ)‖ g(ξ)‖ PUNCT ejde-67 282 6 dξ dξ PROPN ejde-67 282 7 . . PUNCT ejde-67 283 1 ejde-2023/39 ejde-2023/39 VERB ejde-67 283 2 reduction reduction NOUN ejde-67 283 3 principle principle NOUN ejde-67 283 4 11 11 NUM ejde-67 283 5 therefore therefore ADV ejde-67 283 6 , , PUNCT ejde-67 283 7 yk(t yk(t X ejde-67 283 8 ) ) PUNCT ejde-67 283 9 ∈ ∈ PROPN ejde-67 283 10 ap ap PROPN ejde-67 283 11 ( ( PUNCT ejde-67 283 12 r r NOUN ejde-67 283 13 , , PUNCT ejde-67 283 14 x x NOUN ejde-67 283 15 ) ) PUNCT ejde-67 283 16 for for ADP ejde-67 283 17 k k PROPN ejde-67 283 18 ≥ ≥ PROPN ejde-67 283 19 1 1 NUM ejde-67 283 20 . . PUNCT ejde-67 284 1 on on ADP ejde-67 284 2 the the DET ejde-67 284 3 other other ADJ ejde-67 284 4 hand hand NOUN ejde-67 284 5 , , PUNCT ejde-67 284 6 for for ADP ejde-67 284 7 each each DET ejde-67 284 8 t t NOUN ejde-67 284 9 ∈ ∈ PROPN ejde-67 284 10 r r NOUN ejde-67 284 11 and and CCONJ ejde-67 284 12 k k PROPN ejde-67 284 13 ≥ ≥ NUM ejde-67 284 14 1 1 NUM ejde-67 284 15 , , PUNCT ejde-67 284 16 we we PRON ejde-67 284 17 have have VERB ejde-67 284 18 ‖yk(t)‖ ‖yk(t)‖ NUM ejde-67 284 19 ≤mm̃‖πs‖ ≤mm̃‖πs‖ ADJ ejde-67 285 1 ∫ ∫ PROPN ejde-67 286 1 k k PROPN ejde-67 286 2 k−1 k−1 PROPN ejde-67 287 1 e−αξ‖f(t− e−αξ‖f(t− PROPN ejde-67 287 2 ξ)‖ ξ)‖ NOUN ejde-67 287 3 dξ dξ ADP ejde-67 287 4 ≤mm̃‖πs‖e−α(k−1 ≤mm̃‖πs‖e−α(k−1 PROPN ejde-67 287 5 ) ) PUNCT ejde-67 288 1 ∫ ∫ PROPN ejde-67 289 1 k k PROPN ejde-67 289 2 k−1 k−1 PROPN ejde-67 289 3 ‖f(t− ‖f(t− PROPN ejde-67 289 4 ξ)‖ ξ)‖ NOUN ejde-67 289 5 dξ dξ PROPN ejde-67 289 6 = = PROPN ejde-67 289 7 mm̃‖πs‖e−α(k−1 mm̃‖πs‖e−α(k−1 PROPN ejde-67 289 8 ) ) PUNCT ejde-67 289 9 ∫ ∫ PROPN ejde-67 290 1 t−k+1 t−k+1 VERB ejde-67 290 2 t−k t−k PROPN ejde-67 290 3 ‖f(ξ)‖ ‖f(ξ)‖ PUNCT ejde-67 290 4 dξ dξ VERB ejde-67 290 5 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 290 6 ‖f‖bs1e−α(k−1 ‖f‖bs1e−α(k−1 PROPN ejde-67 290 7 ) ) PUNCT ejde-67 290 8 . . PUNCT ejde-67 291 1 from from ADP ejde-67 291 2 the the DET ejde-67 291 3 well well ADV ejde-67 291 4 - - PUNCT ejde-67 291 5 known know VERB ejde-67 291 6 weierstrass weierstrass NOUN ejde-67 291 7 theorem theorem NOUN ejde-67 291 8 we we PRON ejde-67 291 9 deduce deduce VERB ejde-67 291 10 that that SCONJ ejde-67 291 11 the the DET ejde-67 291 12 series series PROPN ejde-67 291 13 ∑∞ ∑∞ PROPN ejde-67 291 14 k=1yk(t k=1yk(t PROPN ejde-67 291 15 ) ) PUNCT ejde-67 291 16 is be AUX ejde-67 291 17 uniformly uniformly ADV ejde-67 291 18 convergent convergent NOUN ejde-67 291 19 on on ADP ejde-67 291 20 r. r. PROPN ejde-67 291 21 let let VERB ejde-67 291 22 h(n h(n PROPN ejde-67 291 23 , , PUNCT ejde-67 291 24 ξ ξ PROPN ejde-67 291 25 , , PUNCT ejde-67 291 26 t t PROPN ejde-67 291 27 ) ) PUNCT ejde-67 291 28 = = PUNCT ejde-67 292 1 v v NUM ejde-67 292 2 s(t− s(t− PROPN ejde-67 292 3 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 292 4 ) ) PUNCT ejde-67 292 5 ) ) PUNCT ejde-67 292 6 . . PUNCT ejde-67 293 1 we we PRON ejde-67 293 2 claim claim VERB ejde-67 293 3 that that SCONJ ejde-67 293 4 y y PROPN ejde-67 293 5 ( ( PUNCT ejde-67 293 6 t t PROPN ejde-67 293 7 ) ) PUNCT ejde-67 293 8 = = NOUN ejde-67 293 9 ∑∞ ∑∞ X ejde-67 293 10 k=1yk(t k=1yk(t PROPN ejde-67 293 11 ) ) PUNCT ejde-67 293 12 . . PUNCT ejde-67 294 1 in in ADP ejde-67 294 2 fact fact NOUN ejde-67 294 3 , , PUNCT ejde-67 294 4 ‖ ‖ PROPN ejde-67 294 5 n∑ n∑ NOUN ejde-67 294 6 k=1 k=1 PROPN ejde-67 294 7 yk(t)− yk(t)− PROPN ejde-67 294 8 y y PROPN ejde-67 294 9 ( ( PUNCT ejde-67 294 10 t)‖ t)‖ NOUN ejde-67 294 11 = = SYM ejde-67 294 12 ‖ ‖ PROPN ejde-67 294 13 n∑ n∑ PROPN ejde-67 294 14 k=1 k=1 PROPN ejde-67 294 15 lim lim PROPN ejde-67 294 16 n→∞ n→∞ NUM ejde-67 294 17 ∫ ∫ PROPN ejde-67 294 18 t−k+1 t−k+1 VERB ejde-67 294 19 t−k t−k PROPN ejde-67 294 20 h(n h(n PROPN ejde-67 294 21 , , PUNCT ejde-67 294 22 ξ ξ PROPN ejde-67 294 23 , , PUNCT ejde-67 294 24 t t PROPN ejde-67 294 25 ) ) PUNCT ejde-67 294 26 dξ dξ PROPN ejde-67 294 27 − − PROPN ejde-67 294 28 lim lim PROPN ejde-67 294 29 n→∞ n→∞ NUM ejde-67 294 30 ∫ ∫ PROPN ejde-67 294 31 t t PROPN ejde-67 294 32 −∞ −∞ ADP ejde-67 294 33 h(n h(n PROPN ejde-67 294 34 , , PUNCT ejde-67 294 35 ξ ξ PROPN ejde-67 294 36 , , PUNCT ejde-67 294 37 t t PROPN ejde-67 294 38 ) ) PUNCT ejde-67 294 39 dξ‖ dξ‖ NOUN ejde-67 294 40 = = SYM ejde-67 294 41 ‖ ‖ PROPN ejde-67 294 42 lim lim PROPN ejde-67 294 43 n→∞ n→∞ NOUN ejde-67 295 1 ∞∑ ∞∑ NUM ejde-67 295 2 k k X ejde-67 295 3 = = PROPN ejde-67 295 4 n+1 n+1 PROPN ejde-67 295 5 ∫ ∫ PROPN ejde-67 295 6 t−k+1 t−k+1 VERB ejde-67 295 7 t−k t−k PROPN ejde-67 295 8 h(n h(n PROPN ejde-67 295 9 , , PUNCT ejde-67 295 10 ξ ξ PROPN ejde-67 295 11 , , PUNCT ejde-67 295 12 t t PROPN ejde-67 295 13 ) ) PUNCT ejde-67 295 14 dξ‖ dξ‖ NOUN ejde-67 295 15 ≤mm̃‖πs‖ ≤mm̃‖πs‖ VERB ejde-67 295 16 ∞∑ ∞∑ NUM ejde-67 295 17 k k NOUN ejde-67 295 18 = = PROPN ejde-67 295 19 n+1 n+1 PROPN ejde-67 295 20 ∫ ∫ PROPN ejde-67 295 21 t−k+1 t−k+1 ADP ejde-67 295 22 t−k t−k PROPN ejde-67 295 23 e−α(t−ξ)‖f(ξ)‖ e−α(t−ξ)‖f(ξ)‖ X ejde-67 295 24 dξ dξ PROPN ejde-67 295 25 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 295 26 ‖f‖bs1 ‖f‖bs1 PROPN ejde-67 296 1 ∞∑ ∞∑ NUM ejde-67 296 2 k k X ejde-67 296 3 = = ADJ ejde-67 296 4 n+1 n+1 NOUN ejde-67 296 5 e−α(k−1 e−α(k−1 NOUN ejde-67 296 6 ) ) PUNCT ejde-67 296 7 → → SYM ejde-67 296 8 0 0 NUM ejde-67 296 9 as as ADP ejde-67 296 10 n n X ejde-67 296 11 →∞. →∞. PROPN ejde-67 296 12 because because SCONJ ejde-67 296 13 the the DET ejde-67 296 14 convergence convergence NOUN ejde-67 296 15 of of ADP ejde-67 296 16 the the DET ejde-67 296 17 series series NOUN ejde-67 296 18 ∑∞ ∑∞ PROPN ejde-67 296 19 k=1yk(t k=1yk(t PROPN ejde-67 296 20 ) ) PUNCT ejde-67 296 21 is be AUX ejde-67 296 22 uniform uniform ADJ ejde-67 296 23 , , PUNCT ejde-67 297 1 y y PROPN ejde-67 297 2 ( ( PUNCT ejde-67 297 3 t t PROPN ejde-67 297 4 ) ) PUNCT ejde-67 297 5 ∈ ∈ PROPN ejde-67 297 6 ap ap PROPN ejde-67 297 7 ( ( PUNCT ejde-67 297 8 r r NOUN ejde-67 297 9 , , PUNCT ejde-67 297 10 x x NOUN ejde-67 297 11 ) ) PUNCT ejde-67 297 12 . . PUNCT ejde-67 298 1 � � PROPN ejde-67 298 2 now now ADV ejde-67 298 3 , , PUNCT ejde-67 298 4 we we PRON ejde-67 298 5 extend extend VERB ejde-67 298 6 the the DET ejde-67 298 7 results result NOUN ejde-67 298 8 above above ADV ejde-67 298 9 to to ADP ejde-67 298 10 the the DET ejde-67 298 11 almost almost ADV ejde-67 298 12 automorphic automorphic ADJ ejde-67 298 13 case case NOUN ejde-67 298 14 . . PUNCT ejde-67 299 1 in in ADP ejde-67 299 2 [ [ X ejde-67 299 3 11 11 NUM ejde-67 299 4 ] ] PUNCT ejde-67 299 5 , , PUNCT ejde-67 299 6 the the DET ejde-67 299 7 authors author NOUN ejde-67 299 8 established establish VERB ejde-67 299 9 the the DET ejde-67 299 10 following following ADJ ejde-67 299 11 result result NOUN ejde-67 299 12 which which PRON ejde-67 299 13 ensuring ensure VERB ejde-67 299 14 the the DET ejde-67 299 15 existence existence NOUN ejde-67 299 16 of of ADP ejde-67 299 17 almost almost ADV ejde-67 299 18 automorphic automorphic ADJ ejde-67 299 19 solutions solution NOUN ejde-67 299 20 to to ADP ejde-67 299 21 ( ( PUNCT ejde-67 299 22 1.3 1.3 NUM ejde-67 299 23 ) ) PUNCT ejde-67 299 24 . . PUNCT ejde-67 300 1 theorem theorem VERB ejde-67 300 2 5.3 5.3 NUM ejde-67 300 3 ( ( PUNCT ejde-67 300 4 [ [ X ejde-67 300 5 11 11 NUM ejde-67 300 6 ] ] NUM ejde-67 300 7 ) ) PUNCT ejde-67 300 8 . . PUNCT ejde-67 301 1 assume assume VERB ejde-67 301 2 that that SCONJ ejde-67 301 3 the the DET ejde-67 301 4 semigroup semigroup NOUN ejde-67 301 5 ( ( PUNCT ejde-67 301 6 t t PROPN ejde-67 301 7 ( ( PUNCT ejde-67 301 8 t))t≥0 t))t≥0 PROPN ejde-67 301 9 is be AUX ejde-67 301 10 compact compact ADJ ejde-67 301 11 and and CCONJ ejde-67 301 12 f f PROPN ejde-67 301 13 is be AUX ejde-67 301 14 an an DET ejde-67 301 15 almost almost ADV ejde-67 301 16 automorphic automorphic ADJ ejde-67 301 17 function function NOUN ejde-67 301 18 . . PUNCT ejde-67 302 1 moreover moreover ADV ejde-67 302 2 , , PUNCT ejde-67 302 3 if if SCONJ ejde-67 302 4 ( ( PUNCT ejde-67 302 5 1.3 1.3 NUM ejde-67 302 6 ) ) PUNCT ejde-67 302 7 has have VERB ejde-67 302 8 a a DET ejde-67 302 9 bounded bound VERB ejde-67 302 10 solution solution NOUN ejde-67 302 11 on on ADP ejde-67 302 12 r+ r+ NOUN ejde-67 302 13 , , PUNCT ejde-67 302 14 then then ADV ejde-67 302 15 it it PRON ejde-67 302 16 has have VERB ejde-67 302 17 an an DET ejde-67 302 18 almost almost ADV ejde-67 302 19 automorphic automorphic ADJ ejde-67 302 20 solution solution NOUN ejde-67 302 21 . . PUNCT ejde-67 303 1 we we PRON ejde-67 303 2 establish establish VERB ejde-67 303 3 the the DET ejde-67 303 4 same same ADJ ejde-67 303 5 result result NOUN ejde-67 303 6 even even ADV ejde-67 303 7 if if SCONJ ejde-67 303 8 the the DET ejde-67 303 9 semigroup semigroup NOUN ejde-67 303 10 is be AUX ejde-67 303 11 not not PART ejde-67 303 12 necessary necessary ADJ ejde-67 303 13 compact compact ADJ ejde-67 303 14 but but CCONJ ejde-67 303 15 the the DET ejde-67 303 16 operator operator NOUN ejde-67 303 17 t t PROPN ejde-67 303 18 ( ( PUNCT ejde-67 303 19 t)l t)l NOUN ejde-67 303 20 is be AUX ejde-67 303 21 compact compact ADJ ejde-67 303 22 for for ADP ejde-67 303 23 all all DET ejde-67 303 24 t t PROPN ejde-67 303 25 > > X ejde-67 303 26 0 0 X ejde-67 303 27 . . PUNCT ejde-67 304 1 theorem theorem NOUN ejde-67 304 2 5.4 5.4 NUM ejde-67 304 3 . . PUNCT ejde-67 305 1 assume assume VERB ejde-67 305 2 that that SCONJ ejde-67 305 3 the the DET ejde-67 305 4 semigroup semigroup NOUN ejde-67 305 5 ( ( PUNCT ejde-67 305 6 t t PROPN ejde-67 305 7 ( ( PUNCT ejde-67 305 8 t))t≥0 t))t≥0 PROPN ejde-67 305 9 is be AUX ejde-67 305 10 exponentially exponentially ADV ejde-67 305 11 stable stable ADJ ejde-67 305 12 and and CCONJ ejde-67 305 13 that that SCONJ ejde-67 305 14 the the DET ejde-67 305 15 operator operator NOUN ejde-67 305 16 t t PROPN ejde-67 305 17 ( ( PUNCT ejde-67 305 18 t)l t)l NOUN ejde-67 305 19 is be AUX ejde-67 305 20 compact compact ADJ ejde-67 305 21 for for ADP ejde-67 305 22 t t PROPN ejde-67 305 23 > > X ejde-67 305 24 0 0 PROPN ejde-67 305 25 . . PUNCT ejde-67 306 1 if if SCONJ ejde-67 306 2 f f PROPN ejde-67 306 3 : : PUNCT ejde-67 306 4 r r X ejde-67 306 5 → → PUNCT ejde-67 306 6 x x X ejde-67 306 7 is be AUX ejde-67 306 8 a a DET ejde-67 306 9 s1 s1 NOUN ejde-67 306 10 - - PUNCT ejde-67 306 11 almost almost ADV ejde-67 306 12 automorphic automorphic ADJ ejde-67 306 13 function function NOUN ejde-67 306 14 and and CCONJ ejde-67 306 15 ( ( PUNCT ejde-67 306 16 1.3 1.3 NUM ejde-67 306 17 ) ) PUNCT ejde-67 306 18 has have VERB ejde-67 306 19 a a DET ejde-67 306 20 bounded bound VERB ejde-67 306 21 solution solution NOUN ejde-67 306 22 on on ADP ejde-67 306 23 r+ r+ NOUN ejde-67 306 24 , , PUNCT ejde-67 306 25 then then ADV ejde-67 306 26 it it PRON ejde-67 306 27 has have VERB ejde-67 306 28 an an DET ejde-67 306 29 almost almost ADV ejde-67 306 30 automorphic automorphic ADJ ejde-67 306 31 solution solution NOUN ejde-67 306 32 . . PUNCT ejde-67 307 1 to to PART ejde-67 307 2 prove prove VERB ejde-67 307 3 this this DET ejde-67 307 4 theorem theorem NOUN ejde-67 307 5 we we PRON ejde-67 307 6 need need VERB ejde-67 307 7 the the DET ejde-67 307 8 following follow VERB ejde-67 307 9 lemma lemma PROPN ejde-67 307 10 . . PUNCT ejde-67 308 1 lemma lemma PROPN ejde-67 308 2 5.5 5.5 NUM ejde-67 308 3 ( ( PUNCT ejde-67 308 4 [ [ X ejde-67 308 5 2 2 NUM ejde-67 308 6 ] ] PUNCT ejde-67 308 7 ) ) PUNCT ejde-67 308 8 . . PUNCT ejde-67 309 1 assume assume VERB ejde-67 309 2 that that SCONJ ejde-67 309 3 g g PROPN ejde-67 309 4 is be AUX ejde-67 309 5 a a DET ejde-67 309 6 s1 s1 NOUN ejde-67 309 7 - - PUNCT ejde-67 309 8 almost almost ADV ejde-67 309 9 automorphic automorphic ADJ ejde-67 309 10 function function NOUN ejde-67 309 11 . . PUNCT ejde-67 310 1 if if SCONJ ejde-67 310 2 ( ( PUNCT ejde-67 310 3 5.1 5.1 NUM ejde-67 310 4 ) ) PUNCT ejde-67 310 5 has have VERB ejde-67 310 6 a a DET ejde-67 310 7 bounded bound VERB ejde-67 310 8 solution solution NOUN ejde-67 310 9 on on ADP ejde-67 310 10 r+ r+ NOUN ejde-67 310 11 then then ADV ejde-67 310 12 it it PRON ejde-67 310 13 admits admit VERB ejde-67 310 14 an an DET ejde-67 310 15 almost almost ADV ejde-67 310 16 automorphic automorphic ADJ ejde-67 310 17 solution solution NOUN ejde-67 310 18 on on ADP ejde-67 310 19 r. r. PROPN ejde-67 310 20 12 12 NUM ejde-67 310 21 m. m. NOUN ejde-67 310 22 el el PROPN ejde-67 310 23 attaouy attaouy PROPN ejde-67 310 24 , , PUNCT ejde-67 310 25 k. k. PROPN ejde-67 310 26 ezzinbi ezzinbi PROPN ejde-67 310 27 , , PUNCT ejde-67 310 28 g. g. PROPN ejde-67 310 29 m. m. PROPN ejde-67 310 30 n’guérékata n’guérékata PROPN ejde-67 310 31 ejde-2023/39 ejde-2023/39 VERB ejde-67 310 32 proof proof NOUN ejde-67 310 33 of of ADP ejde-67 310 34 theorem theorem NOUN ejde-67 310 35 5.4 5.4 NUM ejde-67 310 36 . . PUNCT ejde-67 311 1 let let VERB ejde-67 311 2 x x PRON ejde-67 311 3 be be AUX ejde-67 311 4 the the DET ejde-67 311 5 mild mild ADJ ejde-67 311 6 solution solution NOUN ejde-67 311 7 of of ADP ejde-67 311 8 ( ( PUNCT ejde-67 311 9 1.3 1.3 NUM ejde-67 311 10 ) ) PUNCT ejde-67 311 11 given give VERB ejde-67 311 12 by by ADP ejde-67 311 13 ( ( PUNCT ejde-67 311 14 3.4 3.4 NUM ejde-67 311 15 ) ) PUNCT ejde-67 311 16 . . PUNCT ejde-67 312 1 since since SCONJ ejde-67 312 2 z(t z(t NOUN ejde-67 312 3 ) ) PUNCT ejde-67 312 4 satisfies satisfie NOUN ejde-67 312 5 ( ( PUNCT ejde-67 312 6 3.3 3.3 NUM ejde-67 312 7 ) ) PUNCT ejde-67 312 8 , , PUNCT ejde-67 312 9 it it PRON ejde-67 312 10 follows follow VERB ejde-67 312 11 that that SCONJ ejde-67 312 12 z z NOUN ejde-67 312 13 ( ( PUNCT ejde-67 312 14 · · PUNCT ejde-67 312 15 ) ) PUNCT ejde-67 312 16 is be AUX ejde-67 312 17 bounded bound VERB ejde-67 312 18 on on ADP ejde-67 312 19 r+ r+ X ejde-67 312 20 . . PUNCT ejde-67 313 1 moreover moreover ADV ejde-67 313 2 , , PUNCT ejde-67 313 3 the the DET ejde-67 313 4 function function NOUN ejde-67 313 5 g(t g(t PROPN ejde-67 313 6 ) ) PUNCT ejde-67 314 1 = = PUNCT ejde-67 314 2 〈 〈 NOUN ejde-67 314 3 x∗ x∗ PROPN ejde-67 314 4 , , PUNCT ejde-67 314 5 f(t f(t PROPN ejde-67 314 6 ) ) PUNCT ejde-67 314 7 〉 〉 PROPN ejde-67 314 8 is be AUX ejde-67 314 9 s1 s1 NOUN ejde-67 314 10 - - PUNCT ejde-67 314 11 almost almost ADV ejde-67 314 12 automorphic automorphic ADJ ejde-67 314 13 . . PUNCT ejde-67 315 1 by by ADP ejde-67 315 2 lemma lemma PROPN ejde-67 315 3 5.5 5.5 NUM ejde-67 315 4 , , PUNCT ejde-67 315 5 we we PRON ejde-67 315 6 obtain obtain VERB ejde-67 315 7 that that SCONJ ejde-67 315 8 z z NOUN ejde-67 315 9 ( ( PUNCT ejde-67 315 10 · · PUNCT ejde-67 315 11 ) ) PUNCT ejde-67 315 12 is be AUX ejde-67 315 13 almost almost ADV ejde-67 315 14 automorphic automorphic ADJ ejde-67 315 15 on on ADP ejde-67 315 16 r r NOUN ejde-67 315 17 and and CCONJ ejde-67 315 18 φ φ PROPN ejde-67 315 19 z z PROPN ejde-67 315 20 ( ( PUNCT ejde-67 315 21 · · PUNCT ejde-67 315 22 ) ) PUNCT ejde-67 315 23 is be AUX ejde-67 315 24 an an DET ejde-67 315 25 almost almost ADV ejde-67 315 26 automorphic automorphic ADJ ejde-67 315 27 function function NOUN ejde-67 315 28 on on ADP ejde-67 315 29 r. r. PROPN ejde-67 315 30 let let VERB ejde-67 315 31 y y PROPN ejde-67 315 32 ( ( PUNCT ejde-67 315 33 t t PROPN ejde-67 315 34 ) ) PUNCT ejde-67 316 1 = = PROPN ejde-67 316 2 lim lim PROPN ejde-67 316 3 n→∞ n→∞ NUM ejde-67 317 1 ∫ ∫ PROPN ejde-67 317 2 t t PROPN ejde-67 317 3 −∞ −∞ ADP ejde-67 317 4 v v PROPN ejde-67 317 5 s(t− s(t− PROPN ejde-67 317 6 ξ)πs(b̃nx0f(ξ))dξ ξ)πs(b̃nx0f(ξ))dξ NUM ejde-67 317 7 , , PUNCT ejde-67 317 8 for for ADP ejde-67 317 9 t t PROPN ejde-67 317 10 ∈ ∈ PROPN ejde-67 317 11 r. r. PROPN ejde-67 317 12 ( ( PUNCT ejde-67 317 13 5.4 5.4 NUM ejde-67 317 14 ) ) PUNCT ejde-67 317 15 since since SCONJ ejde-67 317 16 f f PROPN ejde-67 317 17 is be AUX ejde-67 317 18 s1 s1 NOUN ejde-67 317 19 - - PUNCT ejde-67 317 20 almost almost ADV ejde-67 317 21 automorphic automorphic ADJ ejde-67 317 22 , , PUNCT ejde-67 317 23 for for ADP ejde-67 317 24 any any DET ejde-67 317 25 sequence sequence NOUN ejde-67 317 26 of of ADP ejde-67 317 27 real real ADJ ejde-67 317 28 numbers number NOUN ejde-67 317 29 ( ( PUNCT ejde-67 317 30 s′p)p s′p)p X ejde-67 317 31 there there PRON ejde-67 317 32 exists exist VERB ejde-67 317 33 a a DET ejde-67 317 34 subsequence subsequence NOUN ejde-67 317 35 ( ( PUNCT ejde-67 317 36 sp)p sp)p NOUN ejde-67 317 37 of of ADP ejde-67 317 38 ( ( PUNCT ejde-67 317 39 s′p)p s′p)p NOUN ejde-67 317 40 and and CCONJ ejde-67 317 41 a a DET ejde-67 317 42 function function NOUN ejde-67 317 43 g g PROPN ejde-67 317 44 ∈ ∈ PROPN ejde-67 317 45 lploc(r lploc(r NOUN ejde-67 317 46 , , PUNCT ejde-67 317 47 x x NOUN ejde-67 317 48 ) ) PUNCT ejde-67 317 49 such such ADJ ejde-67 317 50 that that SCONJ ejde-67 317 51 , , PUNCT ejde-67 317 52 for for ADP ejde-67 317 53 each each DET ejde-67 317 54 t t NOUN ejde-67 317 55 ∈ ∈ PROPN ejde-67 317 56 r r NOUN ejde-67 317 57 , , PUNCT ejde-67 317 58 sup sup NOUN ejde-67 317 59 t∈r t∈r NOUN ejde-67 317 60 ∫ ∫ NOUN ejde-67 317 61 t+1 t+1 PROPN ejde-67 317 62 t t PROPN ejde-67 317 63 ‖f(s+ ‖f(s+ ADJ ejde-67 317 64 sp)− sp)− PUNCT ejde-67 318 1 g(s)‖ds→ g(s)‖ds→ NOUN ejde-67 318 2 0 0 NUM ejde-67 318 3 as as ADP ejde-67 318 4 p→∞ p→∞ NUM ejde-67 318 5 ( ( PUNCT ejde-67 318 6 5.5 5.5 NUM ejde-67 318 7 ) ) PUNCT ejde-67 318 8 sup sup NOUN ejde-67 318 9 t∈r t∈r NOUN ejde-67 318 10 ∫ ∫ NOUN ejde-67 319 1 t+1 t+1 PROPN ejde-67 319 2 t t PROPN ejde-67 319 3 ‖g(s− ‖g(s− PROPN ejde-67 319 4 sp)− sp)− ADJ ejde-67 319 5 f(s)‖ds→ f(s)‖ds→ NOUN ejde-67 319 6 0 0 NUM ejde-67 319 7 as as ADP ejde-67 319 8 p→∞ p→∞ ADJ ejde-67 319 9 ( ( PUNCT ejde-67 319 10 5.6 5.6 NUM ejde-67 319 11 ) ) PUNCT ejde-67 319 12 on on ADP ejde-67 319 13 the the DET ejde-67 319 14 other other ADJ ejde-67 319 15 hand hand NOUN ejde-67 319 16 , , PUNCT ejde-67 319 17 let let VERB ejde-67 319 18 yk(t yk(t PRON ejde-67 319 19 ) ) PUNCT ejde-67 320 1 = = SYM ejde-67 320 2 lim lim PROPN ejde-67 320 3 n→∞ n→∞ NUM ejde-67 321 1 ∫ ∫ PROPN ejde-67 321 2 k k PROPN ejde-67 321 3 k−1 k−1 PROPN ejde-67 321 4 v v ADP ejde-67 321 5 s(ξ)πs(b̃nx0f(t− s(ξ)πs(b̃nx0f(t− NOUN ejde-67 321 6 ξ))dξ ξ))dξ PROPN ejde-67 321 7 , , PUNCT ejde-67 321 8 for for ADP ejde-67 321 9 t t PROPN ejde-67 321 10 ∈ ∈ PROPN ejde-67 321 11 r r NOUN ejde-67 321 12 , , PUNCT ejde-67 321 13 the the DET ejde-67 321 14 function function NOUN ejde-67 321 15 yk yk NOUN ejde-67 321 16 : : PUNCT ejde-67 321 17 t→ t→ NOUN ejde-67 321 18 yk(t yk(t NUM ejde-67 321 19 ) ) PUNCT ejde-67 321 20 is be AUX ejde-67 321 21 continuous continuous ADJ ejde-67 321 22 on on ADP ejde-67 321 23 r. r. PROPN ejde-67 321 24 let let VERB ejde-67 321 25 zk(t zk(t NOUN ejde-67 321 26 ) ) PUNCT ejde-67 321 27 = = SYM ejde-67 321 28 lim lim PROPN ejde-67 321 29 n→∞ n→∞ NUM ejde-67 322 1 ∫ ∫ PROPN ejde-67 322 2 k k PROPN ejde-67 322 3 k−1 k−1 PROPN ejde-67 322 4 v v PROPN ejde-67 322 5 s(ξ)πs(b̃nx0g(t− s(ξ)πs(b̃nx0g(t− NOUN ejde-67 322 6 ξ))dξ ξ))dξ PROPN ejde-67 322 7 , , PUNCT ejde-67 322 8 for for ADP ejde-67 322 9 t t PROPN ejde-67 322 10 ∈ ∈ PROPN ejde-67 322 11 r r NOUN ejde-67 322 12 , , PUNCT ejde-67 322 13 we we PRON ejde-67 322 14 obtain obtain VERB ejde-67 322 15 that that DET ejde-67 322 16 ‖yk(t+ ‖yk(t+ ADJ ejde-67 322 17 sp)− sp)− ADJ ejde-67 323 1 zk(t)‖ zk(t)‖ NOUN ejde-67 323 2 = = SYM ejde-67 323 3 ‖ ‖ PROPN ejde-67 323 4 lim lim PROPN ejde-67 323 5 n→∞ n→∞ NUM ejde-67 324 1 ∫ ∫ PROPN ejde-67 324 2 k k PROPN ejde-67 324 3 k−1 k−1 PROPN ejde-67 324 4 v v ADP ejde-67 324 5 s(ξ)πs(b̃nx0(f(t+ s(ξ)πs(b̃nx0(f(t+ NOUN ejde-67 324 6 sp sp ADP ejde-67 324 7 − − PROPN ejde-67 324 8 ξ)− ξ)− PROPN ejde-67 324 9 g(t− g(t− PROPN ejde-67 324 10 ξ ξ PROPN ejde-67 324 11 ) ) PUNCT ejde-67 324 12 ) ) PUNCT ejde-67 324 13 ) ) PUNCT ejde-67 325 1 dξ‖ dξ‖ NOUN ejde-67 326 1 ≤mm̃‖πs‖ ≤mm̃‖πs‖ ADJ ejde-67 326 2 ∫ ∫ PROPN ejde-67 327 1 k k PROPN ejde-67 327 2 k−1 k−1 PROPN ejde-67 327 3 e−αξ‖f(t+ e−αξ‖f(t+ PROPN ejde-67 327 4 sp sp ADP ejde-67 327 5 − − PROPN ejde-67 327 6 ξ)− ξ)− PROPN ejde-67 327 7 g(t− g(t− PROPN ejde-67 327 8 ξ)‖ ξ)‖ NOUN ejde-67 327 9 dξ dξ PROPN ejde-67 327 10 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 327 11 e−α(k−1 e−α(k−1 NOUN ejde-67 327 12 ) ) PUNCT ejde-67 328 1 ∫ ∫ PROPN ejde-67 329 1 k k PROPN ejde-67 329 2 k−1 k−1 PROPN ejde-67 329 3 ‖f(t+ ‖f(t+ PROPN ejde-67 329 4 sp sp ADP ejde-67 329 5 − − PROPN ejde-67 329 6 ξ)− ξ)− PROPN ejde-67 329 7 g(t− g(t− PROPN ejde-67 329 8 ξ)‖ ξ)‖ VERB ejde-67 329 9 dξ dξ PROPN ejde-67 330 1 = = PUNCT ejde-67 330 2 mm̃‖πs‖ mm̃‖πs‖ NOUN ejde-67 330 3 e−α(k−1 e−α(k−1 NOUN ejde-67 330 4 ) ) PUNCT ejde-67 330 5 ∫ ∫ PROPN ejde-67 331 1 t−k+1 t−k+1 VERB ejde-67 331 2 t−k t−k NOUN ejde-67 331 3 ‖f(sp ‖f(sp PROPN ejde-67 332 1 + + CCONJ ejde-67 332 2 ξ)− ξ)− PROPN ejde-67 332 3 g(ξ)‖ g(ξ)‖ PUNCT ejde-67 332 4 dξ dξ ADP ejde-67 332 5 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 332 6 e−α(k−1 e−α(k−1 NOUN ejde-67 332 7 ) ) PUNCT ejde-67 332 8 sup sup NOUN ejde-67 332 9 s∈r s∈r PROPN ejde-67 332 10 ∫ ∫ PROPN ejde-67 332 11 s s PART ejde-67 333 1 s+1 s+1 PROPN ejde-67 333 2 ‖f(sp ‖f(sp PROPN ejde-67 333 3 + + CCONJ ejde-67 333 4 ξ)− ξ)− PROPN ejde-67 333 5 g(ξ)‖ g(ξ)‖ PUNCT ejde-67 333 6 dξ dξ PRON ejde-67 333 7 . . PUNCT ejde-67 334 1 using use VERB ejde-67 334 2 the the DET ejde-67 334 3 same same ADJ ejde-67 334 4 argument argument NOUN ejde-67 334 5 as as ADP ejde-67 334 6 above above ADV ejde-67 334 7 , , PUNCT ejde-67 334 8 we we PRON ejde-67 334 9 can can AUX ejde-67 334 10 prove prove VERB ejde-67 334 11 that that SCONJ ejde-67 334 12 zk(t− zk(t− PROPN ejde-67 334 13 sp)→ sp)→ VERB ejde-67 334 14 lim lim PROPN ejde-67 334 15 n→∞ n→∞ NUM ejde-67 335 1 ∫ ∫ PROPN ejde-67 335 2 k k PROPN ejde-67 335 3 k−1 k−1 PROPN ejde-67 335 4 v v INTJ ejde-67 335 5 s(ξ)πs(b̃nx0f(t− s(ξ)πs(b̃nx0f(t− NOUN ejde-67 335 6 ξ))dξ ξ))dξ PROPN ejde-67 335 7 as as ADP ejde-67 335 8 n→∞. n→∞. ADJ ejde-67 335 9 therefore therefore ADV ejde-67 335 10 , , PUNCT ejde-67 335 11 yk(t yk(t X ejde-67 335 12 ) ) PUNCT ejde-67 335 13 ∈ ∈ PROPN ejde-67 335 14 aa(r aa(r NOUN ejde-67 335 15 , , PUNCT ejde-67 335 16 x x NOUN ejde-67 335 17 ) ) PUNCT ejde-67 335 18 for for ADP ejde-67 335 19 k k PROPN ejde-67 335 20 ≥ ≥ PROPN ejde-67 335 21 1 1 NUM ejde-67 335 22 . . PUNCT ejde-67 336 1 on on ADP ejde-67 336 2 other other ADJ ejde-67 336 3 hand hand NOUN ejde-67 336 4 , , PUNCT ejde-67 336 5 for for ADP ejde-67 336 6 each each DET ejde-67 336 7 t t NOUN ejde-67 336 8 ∈ ∈ PROPN ejde-67 336 9 r r NOUN ejde-67 336 10 and and CCONJ ejde-67 336 11 k k PROPN ejde-67 336 12 ≥ ≥ NUM ejde-67 336 13 1 1 NUM ejde-67 336 14 , , PUNCT ejde-67 336 15 we we PRON ejde-67 336 16 have have VERB ejde-67 336 17 ‖yk(t)‖ ‖yk(t)‖ NUM ejde-67 336 18 ≤mm̃‖πs‖ ≤mm̃‖πs‖ ADJ ejde-67 337 1 ∫ ∫ PROPN ejde-67 338 1 k k PROPN ejde-67 338 2 k−1 k−1 PROPN ejde-67 339 1 e−αξ‖f(t− e−αξ‖f(t− PROPN ejde-67 339 2 ξ)‖ ξ)‖ NOUN ejde-67 339 3 dξ dξ ADP ejde-67 339 4 ≤mm̃‖πs‖e−α(k−1 ≤mm̃‖πs‖e−α(k−1 PROPN ejde-67 339 5 ) ) PUNCT ejde-67 340 1 ∫ ∫ PROPN ejde-67 341 1 k k PROPN ejde-67 341 2 k−1 k−1 PROPN ejde-67 341 3 ‖f(t− ‖f(t− PROPN ejde-67 341 4 ξ)‖ ξ)‖ VERB ejde-67 341 5 dξ dξ ADP ejde-67 341 6 ejde-2023/39 ejde-2023/39 ADJ ejde-67 341 7 reduction reduction NOUN ejde-67 341 8 principle principle NOUN ejde-67 341 9 13 13 NUM ejde-67 341 10 = = SYM ejde-67 341 11 mm̃‖πs‖e−α(k−1 mm̃‖πs‖e−α(k−1 PROPN ejde-67 341 12 ) ) PUNCT ejde-67 341 13 ∫ ∫ PROPN ejde-67 342 1 t−k+1 t−k+1 VERB ejde-67 342 2 t−k t−k PROPN ejde-67 342 3 ‖f(ξ)‖ ‖f(ξ)‖ PUNCT ejde-67 342 4 dξ dξ VERB ejde-67 342 5 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 342 6 ‖f‖bs1e−α(k−1 ‖f‖bs1e−α(k−1 PROPN ejde-67 342 7 ) ) PUNCT ejde-67 343 1 we we PRON ejde-67 343 2 deduce deduce VERB ejde-67 343 3 from from ADP ejde-67 343 4 the the DET ejde-67 343 5 well well ADV ejde-67 343 6 - - PUNCT ejde-67 343 7 known know VERB ejde-67 343 8 weierstrass weierstrass NOUN ejde-67 343 9 theorem theorem NOUN ejde-67 343 10 that that SCONJ ejde-67 343 11 the the DET ejde-67 343 12 series series PROPN ejde-67 343 13 ∑∞ ∑∞ PROPN ejde-67 343 14 k=1yk(t k=1yk(t PROPN ejde-67 343 15 ) ) PUNCT ejde-67 343 16 is be AUX ejde-67 343 17 uniformly uniformly ADV ejde-67 343 18 convergent convergent NOUN ejde-67 343 19 on on ADP ejde-67 343 20 r. r. PROPN ejde-67 343 21 let let VERB ejde-67 343 22 h(n h(n PROPN ejde-67 343 23 , , PUNCT ejde-67 343 24 ξ ξ PROPN ejde-67 343 25 , , PUNCT ejde-67 343 26 t t PROPN ejde-67 343 27 ) ) PUNCT ejde-67 343 28 = = PUNCT ejde-67 344 1 v v NUM ejde-67 344 2 s(t− s(t− PROPN ejde-67 344 3 ξ)πs(b̃nx0f(ξ ξ)πs(b̃nx0f(ξ NOUN ejde-67 344 4 ) ) PUNCT ejde-67 344 5 ) ) PUNCT ejde-67 344 6 . . PUNCT ejde-67 345 1 we we PRON ejde-67 345 2 claim claim VERB ejde-67 345 3 that that SCONJ ejde-67 345 4 y y PROPN ejde-67 345 5 ( ( PUNCT ejde-67 345 6 t t PROPN ejde-67 345 7 ) ) PUNCT ejde-67 345 8 = = NOUN ejde-67 345 9 ∑∞ ∑∞ X ejde-67 345 10 k=1yk(t k=1yk(t PROPN ejde-67 345 11 ) ) PUNCT ejde-67 345 12 . . PUNCT ejde-67 346 1 in in ADP ejde-67 346 2 fact fact NOUN ejde-67 346 3 , , PUNCT ejde-67 346 4 ‖ ‖ PROPN ejde-67 346 5 n∑ n∑ NOUN ejde-67 346 6 k=1 k=1 PROPN ejde-67 346 7 yk(t)− yk(t)− PROPN ejde-67 346 8 y y PROPN ejde-67 346 9 ( ( PUNCT ejde-67 346 10 t)‖ t)‖ NOUN ejde-67 346 11 = = SYM ejde-67 346 12 ‖ ‖ PROPN ejde-67 346 13 n∑ n∑ PROPN ejde-67 346 14 k=1 k=1 PROPN ejde-67 346 15 lim lim PROPN ejde-67 346 16 n→∞ n→∞ NUM ejde-67 346 17 ∫ ∫ PROPN ejde-67 346 18 t−k+1 t−k+1 VERB ejde-67 346 19 t−k t−k PROPN ejde-67 346 20 h(n h(n PROPN ejde-67 346 21 , , PUNCT ejde-67 346 22 ξ ξ PROPN ejde-67 346 23 , , PUNCT ejde-67 346 24 t t PROPN ejde-67 346 25 ) ) PUNCT ejde-67 346 26 dξ dξ PROPN ejde-67 346 27 − − PROPN ejde-67 346 28 lim lim PROPN ejde-67 346 29 n→∞ n→∞ NUM ejde-67 346 30 ∫ ∫ PROPN ejde-67 346 31 t t PROPN ejde-67 346 32 −∞ −∞ ADP ejde-67 346 33 h(n h(n PROPN ejde-67 346 34 , , PUNCT ejde-67 346 35 ξ ξ PROPN ejde-67 346 36 , , PUNCT ejde-67 346 37 t t PROPN ejde-67 346 38 ) ) PUNCT ejde-67 346 39 dξ‖ dξ‖ NOUN ejde-67 346 40 = = SYM ejde-67 346 41 ‖ ‖ PROPN ejde-67 346 42 lim lim PROPN ejde-67 346 43 n→∞ n→∞ NOUN ejde-67 347 1 ∞∑ ∞∑ NUM ejde-67 347 2 k k X ejde-67 347 3 = = PROPN ejde-67 347 4 n+1 n+1 PROPN ejde-67 347 5 ∫ ∫ PROPN ejde-67 347 6 t−k+1 t−k+1 VERB ejde-67 347 7 t−k t−k PROPN ejde-67 347 8 h(n h(n PROPN ejde-67 347 9 , , PUNCT ejde-67 347 10 ξ ξ PROPN ejde-67 347 11 , , PUNCT ejde-67 347 12 t t PROPN ejde-67 347 13 ) ) PUNCT ejde-67 347 14 dξ‖ dξ‖ NOUN ejde-67 347 15 ≤mm̃‖πs‖ ≤mm̃‖πs‖ VERB ejde-67 347 16 ∞∑ ∞∑ NUM ejde-67 347 17 k k NOUN ejde-67 347 18 = = PROPN ejde-67 347 19 n+1 n+1 PROPN ejde-67 347 20 ∫ ∫ PROPN ejde-67 347 21 t−k+1 t−k+1 ADP ejde-67 347 22 t−k t−k PROPN ejde-67 347 23 e−α(t−ξ)‖f(ξ)‖ e−α(t−ξ)‖f(ξ)‖ X ejde-67 347 24 dξ dξ PROPN ejde-67 347 25 ≤mm̃‖πs‖ ≤mm̃‖πs‖ PROPN ejde-67 347 26 ‖f‖bs1 ‖f‖bs1 PROPN ejde-67 348 1 ∞∑ ∞∑ NUM ejde-67 348 2 k k X ejde-67 348 3 = = ADJ ejde-67 348 4 n+1 n+1 NOUN ejde-67 348 5 e−α(k−1 e−α(k−1 NOUN ejde-67 348 6 ) ) PUNCT ejde-67 348 7 → → SYM ejde-67 348 8 0 0 NUM ejde-67 348 9 as as ADP ejde-67 348 10 n n X ejde-67 348 11 →∞ →∞ NOUN ejde-67 348 12 because because SCONJ ejde-67 348 13 the the DET ejde-67 348 14 convergence convergence NOUN ejde-67 348 15 of of ADP ejde-67 348 16 the the DET ejde-67 348 17 series series NOUN ejde-67 348 18 ∑∞ ∑∞ PROPN ejde-67 348 19 k=1yk(t k=1yk(t PROPN ejde-67 348 20 ) ) PUNCT ejde-67 348 21 is be AUX ejde-67 348 22 uniform uniform ADJ ejde-67 348 23 , , PUNCT ejde-67 348 24 we we PRON ejde-67 348 25 deduce deduce VERB ejde-67 348 26 that that SCONJ ejde-67 348 27 y y PROPN ejde-67 348 28 ( ( PUNCT ejde-67 348 29 t t PROPN ejde-67 348 30 ) ) PUNCT ejde-67 348 31 ∈ ∈ PROPN ejde-67 348 32 aa(r aa(r PROPN ejde-67 348 33 , , PUNCT ejde-67 348 34 x x NOUN ejde-67 348 35 ) ) PUNCT ejde-67 348 36 . . PUNCT ejde-67 349 1 hence hence ADV ejde-67 349 2 , , PUNCT ejde-67 349 3 the the DET ejde-67 349 4 ( ( PUNCT ejde-67 349 5 1.3 1.3 NUM ejde-67 349 6 ) ) PUNCT ejde-67 349 7 has have VERB ejde-67 349 8 an an DET ejde-67 349 9 almost almost ADV ejde-67 349 10 automorphic automorphic ADJ ejde-67 349 11 solution solution NOUN ejde-67 349 12 on on ADP ejde-67 349 13 r. r. PROPN ejde-67 349 14 � � PROPN ejde-67 349 15 6 6 NUM ejde-67 349 16 . . PUNCT ejde-67 349 17 application application NOUN ejde-67 349 18 to to PART ejde-67 349 19 apply apply VERB ejde-67 349 20 the the DET ejde-67 349 21 abstract abstract ADJ ejde-67 349 22 results result NOUN ejde-67 349 23 of of ADP ejde-67 349 24 the the DET ejde-67 349 25 previous previous ADJ ejde-67 349 26 section section NOUN ejde-67 349 27 , , PUNCT ejde-67 349 28 we we PRON ejde-67 349 29 consider consider VERB ejde-67 349 30 the the DET ejde-67 349 31 following follow VERB ejde-67 349 32 transportation transportation NOUN ejde-67 349 33 equation equation NOUN ejde-67 349 34 with with ADP ejde-67 349 35 delay delay NOUN ejde-67 349 36 proposed propose VERB ejde-67 349 37 in in ADP ejde-67 349 38 [ [ X ejde-67 349 39 12 12 NUM ejde-67 349 40 ] ] X ejde-67 349 41 : : PUNCT ejde-67 349 42 ∂ ∂ NUM ejde-67 349 43 w(t w(t PROPN ejde-67 349 44 , , PUNCT ejde-67 349 45 ξ ξ NOUN ejde-67 349 46 ) ) PUNCT ejde-67 349 47 ∂ ∂ NUM ejde-67 349 48 t t NOUN ejde-67 349 49 + + CCONJ ejde-67 349 50 ∂ ∂ NUM ejde-67 349 51 w(t w(t PROPN ejde-67 349 52 , , PUNCT ejde-67 349 53 ξ ξ X ejde-67 349 54 ) ) PUNCT ejde-67 349 55 ∂ξ ∂ξ NOUN ejde-67 350 1 + + CCONJ ejde-67 350 2 αw(t αw(t ADJ ejde-67 350 3 , , PUNCT ejde-67 350 4 ξ ξ X ejde-67 350 5 ) ) PUNCT ejde-67 350 6 + + NUM ejde-67 350 7 ∫ ∫ PROPN ejde-67 350 8 ∞ ∞ PROPN ejde-67 350 9 −∞ −∞ ADP ejde-67 350 10 g(ξ g(ξ PROPN ejde-67 350 11 , , PUNCT ejde-67 350 12 η)w(t− η)w(t− NOUN ejde-67 350 13 r r PROPN ejde-67 350 14 , , PUNCT ejde-67 350 15 η η NOUN ejde-67 350 16 ) ) PUNCT ejde-67 350 17 dη dη NOUN ejde-67 350 18 = = SYM ejde-67 350 19 f̃(t f̃(t PROPN ejde-67 350 20 , , PUNCT ejde-67 350 21 ξ ξ NOUN ejde-67 350 22 ) ) PUNCT ejde-67 350 23 , , PUNCT ejde-67 350 24 ξ ξ PROPN ejde-67 350 25 ∈ ∈ PROPN ejde-67 350 26 r r PROPN ejde-67 350 27 , , PUNCT ejde-67 350 28 t t PROPN ejde-67 350 29 ≥ ≥ NOUN ejde-67 350 30 0 0 PUNCT ejde-67 350 31 w(θ w(θ PROPN ejde-67 350 32 , , PUNCT ejde-67 350 33 ξ ξ NOUN ejde-67 350 34 ) ) PUNCT ejde-67 350 35 = = SYM ejde-67 350 36 ϕ(θ ϕ(θ PROPN ejde-67 350 37 , , PUNCT ejde-67 350 38 ξ ξ PROPN ejde-67 350 39 ) ) PUNCT ejde-67 350 40 , , PUNCT ejde-67 350 41 ξ ξ PROPN ejde-67 350 42 ∈ ∈ PROPN ejde-67 350 43 r r NOUN ejde-67 350 44 , , PUNCT ejde-67 350 45 −r −r ADJ ejde-67 350 46 ≤ ≤ NUM ejde-67 350 47 θ θ PROPN ejde-67 350 48 ≤ ≤ NUM ejde-67 350 49 0 0 NUM ejde-67 350 50 , , PUNCT ejde-67 350 51 ( ( PUNCT ejde-67 350 52 6.1 6.1 NUM ejde-67 350 53 ) ) PUNCT ejde-67 350 54 where where SCONJ ejde-67 350 55 α α X ejde-67 350 56 , , PUNCT ejde-67 350 57 r r NOUN ejde-67 350 58 > > X ejde-67 350 59 0 0 NUM ejde-67 350 60 and and CCONJ ejde-67 350 61 g g NOUN ejde-67 350 62 , , PUNCT ejde-67 350 63 f̃ f̃ PROPN ejde-67 350 64 , , PUNCT ejde-67 350 65 ϕ ϕ PROPN ejde-67 350 66 are be AUX ejde-67 350 67 continuous continuous ADJ ejde-67 350 68 functions function NOUN ejde-67 350 69 and and CCONJ ejde-67 350 70 the the DET ejde-67 350 71 function function NOUN ejde-67 350 72 f̃ f̃ PROPN ejde-67 350 73 : : PUNCT ejde-67 350 74 r×r→ r×r→ X ejde-67 350 75 r r NOUN ejde-67 350 76 is be AUX ejde-67 350 77 given give VERB ejde-67 350 78 by by ADP ejde-67 350 79 f̃(t f̃(t PROPN ejde-67 350 80 , , PUNCT ejde-67 350 81 ξ ξ X ejde-67 350 82 ) ) PUNCT ejde-67 350 83 = = VERB ejde-67 350 84 sin sin NOUN ejde-67 350 85 ( ( PUNCT ejde-67 350 86 1 1 NUM ejde-67 350 87 2 2 NUM ejde-67 350 88 + + CCONJ ejde-67 350 89 cos cos PROPN ejde-67 350 90 t+ t+ NOUN ejde-67 350 91 cos cos PROPN ejde-67 350 92 √ √ PROPN ejde-67 350 93 2 2 NUM ejde-67 350 94 t t NOUN ejde-67 350 95 ) ) PUNCT ejde-67 350 96 h0(ξ h0(ξ NOUN ejde-67 350 97 ) ) PUNCT ejde-67 350 98 + + CCONJ ejde-67 350 99 a(t a(t NOUN ejde-67 350 100 ) ) PUNCT ejde-67 350 101 ( ( PUNCT ejde-67 350 102 6.2 6.2 NUM ejde-67 350 103 ) ) PUNCT ejde-67 350 104 where where SCONJ ejde-67 350 105 h0 h0 PROPN ejde-67 350 106 ∈ ∈ PROPN ejde-67 350 107 l2(r l2(r PROPN ejde-67 350 108 ) ) PUNCT ejde-67 350 109 and and CCONJ ejde-67 350 110 a(t a(t NOUN ejde-67 350 111 ) ) PUNCT ejde-67 350 112 = = SYM ejde-67 350 113 ∑ ∑ PUNCT ejde-67 350 114 n≥1 n≥1 NOUN ejde-67 350 115 βn(t βn(t PUNCT ejde-67 350 116 ) ) PUNCT ejde-67 350 117 such such ADJ ejde-67 350 118 that that PRON ejde-67 350 119 for for ADP ejde-67 350 120 each each DET ejde-67 350 121 n n PRON ejde-67 350 122 ≥ ≥ NOUN ejde-67 350 123 0 0 NUM ejde-67 350 124 , , PUNCT ejde-67 350 125 βn(t βn(t PUNCT ejde-67 350 126 ) ) PUNCT ejde-67 350 127 = = PUNCT ejde-67 351 1 ∑ ∑ PUNCT ejde-67 351 2 i∈pn i∈pn VERB ejde-67 351 3 h(n2(t− h(n2(t− PROPN ejde-67 351 4 i i PROPN ejde-67 351 5 ) ) PUNCT ejde-67 351 6 ) ) PUNCT ejde-67 351 7 . . PUNCT ejde-67 352 1 where where SCONJ ejde-67 352 2 pn pn PROPN ejde-67 352 3 = = SYM ejde-67 352 4 3n(2z 3n(2z NUM ejde-67 352 5 + + NOUN ejde-67 352 6 1 1 NUM ejde-67 352 7 ) ) PUNCT ejde-67 352 8 and and CCONJ ejde-67 352 9 h h NOUN ejde-67 352 10 ∈ ∈ PROPN ejde-67 352 11 c∞(r c∞(r NOUN ejde-67 352 12 , , PUNCT ejde-67 352 13 r r NOUN ejde-67 352 14 ) ) PUNCT ejde-67 352 15 with with ADP ejde-67 352 16 support support NOUN ejde-67 352 17 in in ADP ejde-67 352 18 ( ( PUNCT ejde-67 352 19 −1 −1 NOUN ejde-67 352 20 2 2 NUM ejde-67 352 21 , , PUNCT ejde-67 352 22 1 1 NUM ejde-67 352 23 2 2 NUM ejde-67 352 24 ) ) PUNCT ejde-67 352 25 such such ADJ ejde-67 352 26 that that SCONJ ejde-67 352 27 h h PROPN ejde-67 352 28 ≥ ≥ NOUN ejde-67 352 29 0 0 NUM ejde-67 352 30 ; ; PUNCT ejde-67 352 31 h(0 h(0 PROPN ejde-67 352 32 ) ) PUNCT ejde-67 352 33 = = SYM ejde-67 352 34 1 1 NUM ejde-67 352 35 , , PUNCT ejde-67 352 36 ∫ ∫ PROPN ejde-67 352 37 1 1 NUM ejde-67 352 38 2 2 NUM ejde-67 352 39 −1 −1 NOUN ejde-67 352 40 2 2 NUM ejde-67 352 41 h(s)ds h(s)ds NOUN ejde-67 352 42 = = NOUN ejde-67 352 43 1 1 NUM ejde-67 352 44 . . PUNCT ejde-67 353 1 lemma lemma PROPN ejde-67 353 2 6.1 6.1 NUM ejde-67 353 3 . . PUNCT ejde-67 354 1 [ [ X ejde-67 354 2 17 17 NUM ejde-67 354 3 ] ] PUNCT ejde-67 354 4 the the DET ejde-67 354 5 function function NOUN ejde-67 354 6 a a DET ejde-67 354 7 ∈ ∈ PROPN ejde-67 354 8 c∞(r c∞(r NOUN ejde-67 354 9 , , PUNCT ejde-67 354 10 r r NOUN ejde-67 354 11 ) ) PUNCT ejde-67 354 12 but but CCONJ ejde-67 354 13 a a DET ejde-67 354 14 /∈ /∈ INTJ ejde-67 354 15 aa(r aa(r NOUN ejde-67 354 16 , , PUNCT ejde-67 354 17 r r NOUN ejde-67 354 18 ) ) PUNCT ejde-67 354 19 since since SCONJ ejde-67 354 20 is be AUX ejde-67 354 21 not not PART ejde-67 354 22 bounded bound VERB ejde-67 354 23 on on ADP ejde-67 354 24 r. r. PROPN ejde-67 354 25 however however ADV ejde-67 354 26 , , PUNCT ejde-67 354 27 a a DET ejde-67 354 28 ∈ ∈ NOUN ejde-67 354 29 aas1(r aas1(r NOUN ejde-67 354 30 , , PUNCT ejde-67 354 31 r r NOUN ejde-67 354 32 ) ) PUNCT ejde-67 354 33 . . PUNCT ejde-67 355 1 14 14 NUM ejde-67 355 2 m. m. NOUN ejde-67 355 3 el el PROPN ejde-67 355 4 attaouy attaouy PROPN ejde-67 355 5 , , PUNCT ejde-67 355 6 k. k. PROPN ejde-67 355 7 ezzinbi ezzinbi PROPN ejde-67 355 8 , , PUNCT ejde-67 355 9 g. g. PROPN ejde-67 355 10 m. m. PROPN ejde-67 355 11 n’guérékata n’guérékata PROPN ejde-67 355 12 ejde-2023/39 ejde-2023/39 VERB ejde-67 355 13 to to PART ejde-67 355 14 rewrite rewrite VERB ejde-67 355 15 system system NOUN ejde-67 355 16 ( ( PUNCT ejde-67 355 17 6.1 6.1 NUM ejde-67 355 18 ) ) PUNCT ejde-67 355 19 in in ADP ejde-67 355 20 the the DET ejde-67 355 21 abstract abstract ADJ ejde-67 355 22 form form NOUN ejde-67 355 23 ( ( PUNCT ejde-67 355 24 2.1 2.1 NUM ejde-67 355 25 ) ) PUNCT ejde-67 355 26 , , PUNCT ejde-67 355 27 we we PRON ejde-67 355 28 introduce introduce VERB ejde-67 355 29 the the DET ejde-67 355 30 space space NOUN ejde-67 355 31 x x PUNCT ejde-67 355 32 = = SYM ejde-67 355 33 l2(r l2(r NOUN ejde-67 355 34 ) ) PUNCT ejde-67 355 35 and and CCONJ ejde-67 355 36 we we PRON ejde-67 355 37 define define VERB ejde-67 355 38 the the DET ejde-67 355 39 operator operator NOUN ejde-67 355 40 a a PRON ejde-67 355 41 by by ADP ejde-67 355 42 az(ξ az(ξ NOUN ejde-67 355 43 ) ) PUNCT ejde-67 355 44 = = PRON ejde-67 355 45 −dz(ξ −dz(ξ VERB ejde-67 355 46 ) ) PUNCT ejde-67 355 47 dξ dξ ADP ejde-67 355 48 − − NOUN ejde-67 355 49 αz(ξ αz(ξ NOUN ejde-67 355 50 ) ) PUNCT ejde-67 355 51 on on ADP ejde-67 355 52 the the DET ejde-67 355 53 domain domain NOUN ejde-67 355 54 d(a d(a PROPN ejde-67 355 55 ) ) PUNCT ejde-67 355 56 = = SYM ejde-67 355 57 h1(r h1(r PROPN ejde-67 355 58 ) ) PUNCT ejde-67 355 59 . . PUNCT ejde-67 356 1 from from ADP ejde-67 356 2 [ [ X ejde-67 356 3 12 12 NUM ejde-67 356 4 ] ] PUNCT ejde-67 356 5 we we PRON ejde-67 356 6 obtain obtain VERB ejde-67 356 7 operator operator NOUN ejde-67 356 8 a a PRON ejde-67 356 9 is be AUX ejde-67 356 10 the the DET ejde-67 356 11 infinitesimal infinitesimal ADJ ejde-67 356 12 generator generator NOUN ejde-67 356 13 of of ADP ejde-67 356 14 a a DET ejde-67 356 15 strongly strongly ADV ejde-67 356 16 continuous continuous ADJ ejde-67 356 17 group group NOUN ejde-67 356 18 ( ( PUNCT ejde-67 356 19 t t PROPN ejde-67 356 20 ( ( PUNCT ejde-67 356 21 t))t≥0 t))t≥0 NOUN ejde-67 356 22 on on ADP ejde-67 356 23 x x PUNCT ejde-67 356 24 given give VERB ejde-67 356 25 by by ADP ejde-67 356 26 t t PROPN ejde-67 356 27 ( ( PUNCT ejde-67 356 28 t)z(ξ t)z(ξ ADV ejde-67 356 29 ) ) PUNCT ejde-67 356 30 = = SYM ejde-67 356 31 e−αtz(ξ e−αtz(ξ NUM ejde-67 356 32 − − PROPN ejde-67 356 33 t t PROPN ejde-67 356 34 ) ) PUNCT ejde-67 356 35 , , PUNCT ejde-67 356 36 for for ADP ejde-67 356 37 t t PROPN ejde-67 356 38 ≥ ≥ NOUN ejde-67 356 39 0 0 NUM ejde-67 356 40 and and CCONJ ejde-67 356 41 ξ ξ PROPN ejde-67 356 42 ∈ ∈ PROPN ejde-67 356 43 r. r. PROPN ejde-67 356 44 ( ( PUNCT ejde-67 356 45 6.3 6.3 NUM ejde-67 356 46 ) ) PUNCT ejde-67 356 47 hence hence ADV ejde-67 356 48 , , PUNCT ejde-67 356 49 the the DET ejde-67 356 50 semigroup semigroup NOUN ejde-67 356 51 ( ( PUNCT ejde-67 356 52 t t PROPN ejde-67 356 53 ( ( PUNCT ejde-67 356 54 t))t≥0 t))t≥0 PROPN ejde-67 356 55 , , PUNCT ejde-67 356 56 is be AUX ejde-67 356 57 exponentially exponentially ADV ejde-67 356 58 stable stable ADJ ejde-67 356 59 , , PUNCT ejde-67 356 60 and and CCONJ ejde-67 356 61 the the DET ejde-67 356 62 operator operator NOUN ejde-67 356 63 t t PROPN ejde-67 356 64 ( ( PUNCT ejde-67 356 65 t t PROPN ejde-67 356 66 ) ) PUNCT ejde-67 356 67 is be AUX ejde-67 356 68 not not PART ejde-67 356 69 compact compact ADJ ejde-67 356 70 because because SCONJ ejde-67 356 71 t t PROPN ejde-67 356 72 ( ( PUNCT ejde-67 356 73 t t PROPN ejde-67 356 74 ) ) PUNCT ejde-67 356 75 has have AUX ejde-67 356 76 bounded bound VERB ejde-67 356 77 inverse inverse NOUN ejde-67 356 78 t t PROPN ejde-67 356 79 ( ( PUNCT ejde-67 356 80 −t −t NOUN ejde-67 356 81 ) ) PUNCT ejde-67 356 82 . . PUNCT ejde-67 357 1 let let VERB ejde-67 357 2 f f NOUN ejde-67 357 3 : : PUNCT ejde-67 357 4 r r X ejde-67 357 5 → → PUNCT ejde-67 357 6 x x X ejde-67 357 7 be be VERB ejde-67 357 8 f(t f(t NOUN ejde-67 357 9 ) ) PUNCT ejde-67 357 10 = = SYM ejde-67 357 11 f̃(t f̃(t PROPN ejde-67 357 12 , , PUNCT ejde-67 357 13 · · PUNCT ejde-67 357 14 ) ) PUNCT ejde-67 357 15 , , PUNCT ejde-67 357 16 then then ADV ejde-67 357 17 f f PROPN ejde-67 357 18 is be AUX ejde-67 357 19 a a DET ejde-67 357 20 continuous continuous ADJ ejde-67 357 21 function function NOUN ejde-67 357 22 . . PUNCT ejde-67 358 1 we we PRON ejde-67 358 2 assume assume VERB ejde-67 358 3 that that SCONJ ejde-67 358 4 g g NOUN ejde-67 358 5 : : PUNCT ejde-67 358 6 r2 r2 PROPN ejde-67 359 1 → → PUNCT ejde-67 359 2 r r NOUN ejde-67 359 3 is be AUX ejde-67 359 4 continuous continuous ADJ ejde-67 359 5 and∫ and∫ INTJ ejde-67 359 6 ∞ ∞ PROPN ejde-67 359 7 −∞ −∞ ADP ejde-67 359 8 ∫ ∫ PROPN ejde-67 359 9 ∞ ∞ PROPN ejde-67 359 10 −∞ −∞ PROPN ejde-67 359 11 |g(ξ |g(ξ NOUN ejde-67 359 12 , , PUNCT ejde-67 359 13 η)|2 η)|2 NOUN ejde-67 359 14 dη dη NOUN ejde-67 359 15 dξ dξ ADP ejde-67 359 16 < < X ejde-67 359 17 ∞. ∞. PROPN ejde-67 359 18 lemma lemma PROPN ejde-67 359 19 6.2 6.2 NUM ejde-67 359 20 ( ( PUNCT ejde-67 359 21 [ [ X ejde-67 359 22 12 12 NUM ejde-67 359 23 ] ] PUNCT ejde-67 359 24 ) ) PUNCT ejde-67 359 25 . . PUNCT ejde-67 360 1 under under ADP ejde-67 360 2 the the DET ejde-67 360 3 above above ADJ ejde-67 360 4 conditions condition NOUN ejde-67 360 5 , , PUNCT ejde-67 360 6 the the DET ejde-67 360 7 linear linear ADJ ejde-67 360 8 operator operator NOUN ejde-67 360 9 n n NOUN ejde-67 360 10 : : PUNCT ejde-67 360 11 x x X ejde-67 360 12 → → PUNCT ejde-67 360 13 x x PUNCT ejde-67 360 14 given give VERB ejde-67 360 15 by by ADP ejde-67 360 16 nz(ξ nz(ξ NOUN ejde-67 360 17 ) ) PUNCT ejde-67 360 18 = = SYM ejde-67 361 1 ∫ ∫ PROPN ejde-67 361 2 ∞ ∞ PROPN ejde-67 361 3 −∞ −∞ ADP ejde-67 361 4 g(ξ g(ξ PROPN ejde-67 361 5 , , PUNCT ejde-67 361 6 η)z(η η)z(η NOUN ejde-67 361 7 ) ) PUNCT ejde-67 361 8 dη dη NOUN ejde-67 361 9 is be AUX ejde-67 361 10 compact compact ADJ ejde-67 361 11 . . PUNCT ejde-67 362 1 let let VERB ejde-67 363 1 l l NOUN ejde-67 363 2 : : PUNCT ejde-67 363 3 c([−r c([−r ADJ ejde-67 363 4 , , PUNCT ejde-67 363 5 0 0 NUM ejde-67 363 6 ] ] PUNCT ejde-67 363 7 , , PUNCT ejde-67 363 8 x x X ejde-67 363 9 ) ) PUNCT ejde-67 363 10 → → SYM ejde-67 363 11 x x SYM ejde-67 363 12 defined define VERB ejde-67 363 13 by by ADP ejde-67 363 14 l(ψ l(ψ PROPN ejde-67 363 15 ) ) PUNCT ejde-67 363 16 = = PUNCT ejde-67 363 17 −nψ(−r −nψ(−r NOUN ejde-67 363 18 ) ) PUNCT ejde-67 363 19 . . PUNCT ejde-67 364 1 by by ADP ejde-67 364 2 lemma lemma PROPN ejde-67 364 3 6.2 6.2 NUM ejde-67 364 4 , , PUNCT ejde-67 364 5 we we PRON ejde-67 364 6 obtain obtain VERB ejde-67 364 7 that that SCONJ ejde-67 364 8 l l NOUN ejde-67 364 9 is be AUX ejde-67 364 10 a a DET ejde-67 364 11 compact compact ADJ ejde-67 364 12 linear linear NOUN ejde-67 364 13 map map NOUN ejde-67 364 14 . . PUNCT ejde-67 365 1 with with ADP ejde-67 365 2 this this DET ejde-67 365 3 construction construction NOUN ejde-67 365 4 and and CCONJ ejde-67 365 5 by by ADP ejde-67 365 6 using use VERB ejde-67 365 7 the the DET ejde-67 365 8 notation notation NOUN ejde-67 365 9 x(t x(t PROPN ejde-67 365 10 ) ) PUNCT ejde-67 365 11 = = SYM ejde-67 365 12 w(t w(t PROPN ejde-67 365 13 , , PUNCT ejde-67 365 14 · · PUNCT ejde-67 365 15 ) ) PUNCT ejde-67 365 16 , , PUNCT ejde-67 365 17 the the DET ejde-67 365 18 original original ADJ ejde-67 365 19 system system NOUN ejde-67 365 20 ( ( PUNCT ejde-67 365 21 6.1 6.1 NUM ejde-67 365 22 ) ) PUNCT ejde-67 365 23 is be AUX ejde-67 365 24 represented represent VERB ejde-67 365 25 by by ADP ejde-67 365 26 the the DET ejde-67 365 27 abstract abstract ADJ ejde-67 365 28 system system NOUN ejde-67 365 29 the the DET ejde-67 365 30 abstract abstract ADJ ejde-67 365 31 form form NOUN ejde-67 365 32 ( ( PUNCT ejde-67 365 33 2.1 2.1 NUM ejde-67 365 34 ) ) PUNCT ejde-67 365 35 . . PUNCT ejde-67 366 1 from from ADP ejde-67 366 2 ( ( PUNCT ejde-67 366 3 6.2 6.2 NUM ejde-67 366 4 ) ) PUNCT ejde-67 366 5 we we PRON ejde-67 366 6 obtain obtain VERB ejde-67 366 7 that that SCONJ ejde-67 366 8 f f PROPN ejde-67 366 9 is be AUX ejde-67 366 10 a a DET ejde-67 366 11 s1 s1 NOUN ejde-67 366 12 - - PUNCT ejde-67 366 13 automorphic automorphic ADJ ejde-67 366 14 function function NOUN ejde-67 366 15 . . PUNCT ejde-67 367 1 moreover moreover ADV ejde-67 367 2 the the DET ejde-67 367 3 semigroup semigroup NOUN ejde-67 367 4 ( ( PUNCT ejde-67 367 5 t t PROPN ejde-67 367 6 ( ( PUNCT ejde-67 367 7 t))t≥0 t))t≥0 NOUN ejde-67 367 8 given give VERB ejde-67 367 9 by by ADP ejde-67 367 10 ( ( PUNCT ejde-67 367 11 6.3 6.3 NUM ejde-67 367 12 ) ) PUNCT ejde-67 367 13 is be AUX ejde-67 367 14 exponentially exponentially ADV ejde-67 367 15 stable stable ADJ ejde-67 367 16 and and CCONJ ejde-67 367 17 l l NOUN ejde-67 367 18 is be AUX ejde-67 367 19 a a DET ejde-67 367 20 compact compact ADJ ejde-67 367 21 operator operator NOUN ejde-67 367 22 then then ADV ejde-67 367 23 t t PROPN ejde-67 367 24 ( ( PUNCT ejde-67 367 25 t)l t)l NOUN ejde-67 367 26 is be AUX ejde-67 367 27 compact compact ADJ ejde-67 367 28 . . PUNCT ejde-67 368 1 it it PRON ejde-67 368 2 remains remain VERB ejde-67 368 3 to to PART ejde-67 368 4 show show VERB ejde-67 368 5 that that SCONJ ejde-67 368 6 the the DET ejde-67 368 7 ( ( PUNCT ejde-67 368 8 6.1 6.1 NUM ejde-67 368 9 ) ) PUNCT ejde-67 368 10 has have AUX ejde-67 368 11 bounded bound VERB ejde-67 368 12 mild mild ADJ ejde-67 368 13 solution solution NOUN ejde-67 368 14 on on ADP ejde-67 368 15 r+ r+ X ejde-67 368 16 . . PUNCT ejde-67 369 1 to to PART ejde-67 369 2 achieve achieve VERB ejde-67 369 3 this this DET ejde-67 369 4 goal goal NOUN ejde-67 369 5 , , PUNCT ejde-67 369 6 we we PRON ejde-67 369 7 need need VERB ejde-67 369 8 the the DET ejde-67 369 9 following follow VERB ejde-67 369 10 lemma lemma PROPN ejde-67 369 11 . . PUNCT ejde-67 370 1 lemma lemma PROPN ejde-67 370 2 6.3 6.3 NUM ejde-67 370 3 ( ( PUNCT ejde-67 370 4 [ [ X ejde-67 370 5 7 7 NUM ejde-67 370 6 ] ] NUM ejde-67 370 7 ) ) PUNCT ejde-67 370 8 . . PUNCT ejde-67 371 1 if if SCONJ ejde-67 371 2 x(t x(t PROPN ejde-67 371 3 ) ) PUNCT ejde-67 371 4 ≤ ≤ NOUN ejde-67 371 5 h(t h(t PROPN ejde-67 371 6 ) ) PUNCT ejde-67 372 1 + + CCONJ ejde-67 372 2 ∫ ∫ PROPN ejde-67 372 3 t t PROPN ejde-67 372 4 t0 t0 PROPN ejde-67 372 5 k(s)x(s k(s)x(s PROPN ejde-67 372 6 ) ) PUNCT ejde-67 372 7 ds ds NOUN ejde-67 372 8 for for ADP ejde-67 372 9 [ [ X ejde-67 372 10 t0 t0 PROPN ejde-67 372 11 , , PUNCT ejde-67 372 12 τ τ X ejde-67 372 13 ) ) PUNCT ejde-67 372 14 where where SCONJ ejde-67 372 15 all all PRON ejde-67 372 16 of of ADP ejde-67 372 17 the the DET ejde-67 372 18 functions function NOUN ejde-67 372 19 involved involve VERB ejde-67 372 20 are be AUX ejde-67 372 21 continuous continuous ADJ ejde-67 372 22 and and CCONJ ejde-67 372 23 nonnegative nonnegative VERB ejde-67 372 24 on on ADP ejde-67 372 25 [ [ X ejde-67 372 26 t0 t0 PROPN ejde-67 372 27 , , PUNCT ejde-67 372 28 τ τ X ejde-67 372 29 ) ) PUNCT ejde-67 372 30 and and CCONJ ejde-67 372 31 k(x k(x PROPN ejde-67 372 32 ) ) PUNCT ejde-67 372 33 ≥ ≥ NOUN ejde-67 372 34 0 0 NUM ejde-67 372 35 , , PUNCT ejde-67 372 36 then then ADV ejde-67 372 37 x x AUX ejde-67 372 38 satisfies satisfy VERB ejde-67 372 39 x(t x(t PROPN ejde-67 372 40 ) ) PUNCT ejde-67 372 41 ≤ ≤ NOUN ejde-67 372 42 h(t h(t PROPN ejde-67 372 43 ) ) PUNCT ejde-67 373 1 + + CCONJ ejde-67 373 2 ∫ ∫ PROPN ejde-67 373 3 t t PROPN ejde-67 373 4 t0 t0 PROPN ejde-67 373 5 h(s)k(s)e h(s)k(s)e AUX ejde-67 373 6 ∫ ∫ PROPN ejde-67 373 7 t t PROPN ejde-67 373 8 s s PART ejde-67 373 9 k(u)du k(u)du X ejde-67 373 10 ds ds NOUN ejde-67 373 11 for for ADP ejde-67 373 12 [ [ X ejde-67 373 13 t0 t0 PROPN ejde-67 373 14 , , PUNCT ejde-67 373 15 τ τ X ejde-67 373 16 ) ) PUNCT ejde-67 373 17 for for ADP ejde-67 373 18 any any DET ejde-67 373 19 initial initial ADJ ejde-67 373 20 data datum NOUN ejde-67 373 21 ϕ ϕ PROPN ejde-67 373 22 ∈ ∈ PROPN ejde-67 373 23 c c PROPN ejde-67 373 24 , , PUNCT ejde-67 373 25 the the DET ejde-67 373 26 ( ( PUNCT ejde-67 373 27 6.1 6.1 NUM ejde-67 373 28 ) ) PUNCT ejde-67 373 29 has have VERB ejde-67 373 30 a a DET ejde-67 373 31 solution solution NOUN ejde-67 373 32 x x NOUN ejde-67 373 33 , , PUNCT ejde-67 373 34 given give VERB ejde-67 373 35 by by ADP ejde-67 373 36 x(t x(t PROPN ejde-67 373 37 ) ) PUNCT ejde-67 373 38 = = SYM ejde-67 373 39 t t PROPN ejde-67 373 40 ( ( PUNCT ejde-67 373 41 t)ϕ(0 t)ϕ(0 PROPN ejde-67 373 42 ) ) PUNCT ejde-67 374 1 + + CCONJ ejde-67 374 2 ∫ ∫ PROPN ejde-67 374 3 t t PROPN ejde-67 374 4 0 0 NUM ejde-67 374 5 t t PROPN ejde-67 374 6 ( ( PUNCT ejde-67 374 7 t− t− PROPN ejde-67 374 8 s)(l(xs s)(l(xs PROPN ejde-67 374 9 ) ) PUNCT ejde-67 374 10 + + NUM ejde-67 374 11 f(s f( NOUN ejde-67 374 12 ) ) PUNCT ejde-67 374 13 ) ) PUNCT ejde-67 375 1 ds ds PROPN ejde-67 375 2 , , PUNCT ejde-67 375 3 for for ADP ejde-67 375 4 t t PROPN ejde-67 375 5 ≥ ≥ NOUN ejde-67 375 6 0 0 NUM ejde-67 375 7 . . PUNCT ejde-67 376 1 then then ADV ejde-67 376 2 eαt‖x(t)‖ eαt‖x(t)‖ VERB ejde-67 376 3 ≤ ≤ NUM ejde-67 376 4 ‖ϕ‖+ ‖ϕ‖+ NUM ejde-67 376 5 ∫ ∫ PROPN ejde-67 376 6 t t PROPN ejde-67 376 7 0 0 NUM ejde-67 376 8 eαs(‖l‖‖xs‖+ eαs(‖l‖‖xs‖+ PROPN ejde-67 376 9 ‖f(s)‖ ‖f(s)‖ NOUN ejde-67 376 10 ) ) PUNCT ejde-67 376 11 ds ds PROPN ejde-67 376 12 t t PROPN ejde-67 376 13 ≥ ≥ NOUN ejde-67 376 14 0 0 NUM ejde-67 376 15 . . PUNCT ejde-67 377 1 ( ( PUNCT ejde-67 377 2 6.4 6.4 NUM ejde-67 377 3 ) ) PUNCT ejde-67 377 4 let let VERB ejde-67 377 5 θ θ PROPN ejde-67 377 6 ∈ ∈ PROPN ejde-67 377 7 [ [ X ejde-67 377 8 −r −r ADJ ejde-67 377 9 , , PUNCT ejde-67 377 10 0 0 NUM ejde-67 377 11 ] ] PUNCT ejde-67 377 12 and and CCONJ ejde-67 377 13 t t X ejde-67 377 14 ≥ ≥ NUM ejde-67 377 15 0 0 NUM ejde-67 377 16 . . PUNCT ejde-67 378 1 if if SCONJ ejde-67 378 2 t+ t+ VERB ejde-67 378 3 θ θ PROPN ejde-67 378 4 < < X ejde-67 378 5 0 0 NUM ejde-67 378 6 , , PUNCT ejde-67 378 7 then then ADV ejde-67 378 8 eαt‖x(t+ eαt‖x(t+ VERB ejde-67 378 9 θ)‖ θ)‖ ADV ejde-67 378 10 = = PUNCT ejde-67 378 11 eαt‖ϕ(t+ eαt‖ϕ(t+ NOUN ejde-67 378 12 θ)‖ θ)‖ ADV ejde-67 378 13 ≤ ≤ ADJ ejde-67 378 14 eαr‖ϕ‖ eαr‖ϕ‖ PRON ejde-67 378 15 ≤ ≤ ADV ejde-67 378 16 eαr‖ϕ‖ eαr‖ϕ‖ DET ejde-67 378 17 ejde-2023/39 ejde-2023/39 ADJ ejde-67 378 18 reduction reduction NOUN ejde-67 378 19 principle principle NOUN ejde-67 378 20 15 15 NUM ejde-67 378 21 if if SCONJ ejde-67 378 22 t+ t+ VERB ejde-67 378 23 θ θ PROPN ejde-67 378 24 ≥ ≥ NUM ejde-67 378 25 0 0 NUM ejde-67 378 26 . . PUNCT ejde-67 379 1 by by ADP ejde-67 379 2 using use VERB ejde-67 379 3 ( ( PUNCT ejde-67 379 4 6.4 6.4 NUM ejde-67 379 5 ) ) PUNCT ejde-67 379 6 and and CCONJ ejde-67 379 7 −θ −θ ADJ ejde-67 379 8 ≤ ≤ ADJ ejde-67 379 9 r r NOUN ejde-67 379 10 , , PUNCT ejde-67 379 11 we we PRON ejde-67 379 12 have have AUX ejde-67 379 13 eαt‖x(t+ eαt‖x(t+ NOUN ejde-67 379 14 θ)‖ θ)‖ ADV ejde-67 379 15 ≤ ≤ NOUN ejde-67 379 16 eαr‖ϕ‖+ eαr‖ϕ‖+ ADJ ejde-67 379 17 eαr eαr ADP ejde-67 379 18 ∫ ∫ PROPN ejde-67 379 19 t t PROPN ejde-67 379 20 0 0 NUM ejde-67 379 21 eαs‖f(s)‖ds+ eαs‖f(s)‖ds+ PROPN ejde-67 380 1 eαr‖l‖ eαr‖l‖ PROPN ejde-67 380 2 ∫ ∫ PROPN ejde-67 380 3 t t PROPN ejde-67 380 4 0 0 NUM ejde-67 380 5 eαs‖xs‖ds eαs‖xs‖ds PROPN ejde-67 380 6 . . PUNCT ejde-67 381 1 for for ADP ejde-67 381 2 t t PROPN ejde-67 381 3 ≥ ≥ NOUN ejde-67 381 4 0 0 NUM ejde-67 381 5 let let VERB ejde-67 381 6 eαt‖xt‖ eαt‖xt‖ NOUN ejde-67 381 7 = = PUNCT ejde-67 382 1 sup−r≤θ≤0 sup−r≤θ≤0 NOUN ejde-67 382 2 e e X ejde-67 382 3 αt‖x(t+ αt‖x(t+ PROPN ejde-67 382 4 θ)‖. θ)‖. NOUN ejde-67 382 5 then then ADV ejde-67 382 6 eαt‖xt‖ eαt‖xt‖ VERB ejde-67 382 7 ≤ ≤ PROPN ejde-67 382 8 eαr‖ϕ‖+m2 eαr‖ϕ‖+m2 PROPN ejde-67 382 9 ( ( PUNCT ejde-67 382 10 eα(t+1 eα(t+1 NUM ejde-67 382 11 ) ) PUNCT ejde-67 382 12 − − PROPN ejde-67 383 1 1 1 X ejde-67 383 2 ) ) PUNCT ejde-67 383 3 + + CCONJ ejde-67 384 1 eαr‖l‖ eαr‖l‖ X ejde-67 384 2 ∫ ∫ PROPN ejde-67 384 3 t t PROPN ejde-67 384 4 0 0 NUM ejde-67 384 5 eαs‖xs‖ds eαs‖xs‖ds PROPN ejde-67 384 6 , , PUNCT ejde-67 384 7 where where SCONJ ejde-67 384 8 m2 m2 PROPN ejde-67 384 9 = = SYM ejde-67 384 10 eα(r+1)‖f‖bs1 eα(r+1)‖f‖bs1 PROPN ejde-67 384 11 eα eα VERB ejde-67 384 12 − − PROPN ejde-67 384 13 1 1 NUM ejde-67 384 14 by by ADP ejde-67 384 15 lemma lemma PROPN ejde-67 384 16 6.3 6.3 NUM ejde-67 384 17 we we PRON ejde-67 384 18 obtain obtain VERB ejde-67 384 19 eαt‖xt‖ eαt‖xt‖ ADJ ejde-67 384 20 ≤ ≤ ADJ ejde-67 384 21 eαr‖ϕ‖+m2 eαr‖ϕ‖+m2 PROPN ejde-67 384 22 ( ( PUNCT ejde-67 384 23 eα(t+1 eα(t+1 NUM ejde-67 384 24 ) ) PUNCT ejde-67 384 25 − − PROPN ejde-67 385 1 1 1 X ejde-67 385 2 ) ) PUNCT ejde-67 385 3 + + CCONJ ejde-67 385 4 eαr‖l‖ eαr‖l‖ X ejde-67 385 5 ∫ ∫ PROPN ejde-67 385 6 t t PROPN ejde-67 385 7 0 0 NUM ejde-67 385 8 ( ( PUNCT ejde-67 385 9 eαr‖ϕ‖ eαr‖ϕ‖ PROPN ejde-67 386 1 + + ADJ ejde-67 386 2 m2(eα(s+1 m2(eα(s+1 ADJ ejde-67 386 3 ) ) PUNCT ejde-67 386 4 − − PROPN ejde-67 386 5 1))ee 1))ee NUM ejde-67 386 6 αr‖l‖(t−s αr‖l‖(t− NOUN ejde-67 386 7 ) ) PUNCT ejde-67 386 8 ds ds NOUN ejde-67 386 9 . . PUNCT ejde-67 387 1 moreover moreover ADV ejde-67 387 2 , , PUNCT ejde-67 387 3 if if SCONJ ejde-67 387 4 we we PRON ejde-67 387 5 assume assume VERB ejde-67 387 6 that that SCONJ ejde-67 387 7 ‖l‖ ‖l‖ PROPN ejde-67 387 8 ≤ ≤ NOUN ejde-67 387 9 α α DET ejde-67 387 10 erα erα NOUN ejde-67 387 11 , , PUNCT ejde-67 387 12 then then ADV ejde-67 387 13 ‖xt‖ ‖xt‖ VERB ejde-67 387 14 ≤ ≤ NUM ejde-67 388 1 m2e m2e VERB ejde-67 389 1 α α NOUN ejde-67 389 2 + + X ejde-67 389 3 eαr‖ϕ‖+ eαr‖ϕ‖+ PROPN ejde-67 389 4 m2‖l‖eα(r+1 m2‖l‖eα(r+1 NOUN ejde-67 389 5 ) ) PUNCT ejde-67 389 6 α− α− ADP ejde-67 389 7 ‖l‖eαr ‖l‖eαr NOUN ejde-67 389 8 this this PRON ejde-67 389 9 shows show VERB ejde-67 389 10 that that SCONJ ejde-67 389 11 x x PRON ejde-67 389 12 is be AUX ejde-67 389 13 a a DET ejde-67 389 14 bounded bounded ADJ ejde-67 389 15 solution solution NOUN ejde-67 389 16 of of ADP ejde-67 389 17 ( ( PUNCT ejde-67 389 18 6.1 6.1 NUM ejde-67 389 19 ) ) PUNCT ejde-67 389 20 on on ADP ejde-67 389 21 r+ r+ X ejde-67 389 22 . . PUNCT ejde-67 390 1 as as ADP ejde-67 390 2 a a DET ejde-67 390 3 consequence consequence NOUN ejde-67 390 4 of of ADP ejde-67 390 5 theorem theorem NOUN ejde-67 390 6 5.4 5.4 NUM ejde-67 390 7 , , PUNCT ejde-67 390 8 we we PRON ejde-67 390 9 obtain obtain VERB ejde-67 390 10 that that SCONJ ejde-67 390 11 the the DET ejde-67 390 12 ( ( PUNCT ejde-67 390 13 6.1 6.1 NUM ejde-67 390 14 ) ) PUNCT ejde-67 390 15 has have VERB ejde-67 390 16 an an DET ejde-67 390 17 almost almost ADV ejde-67 390 18 automorphic automorphic ADJ ejde-67 390 19 solution solution NOUN ejde-67 390 20 . . PUNCT ejde-67 391 1 7 7 X ejde-67 391 2 . . X ejde-67 391 3 conclusions conclusion NOUN ejde-67 391 4 and and CCONJ ejde-67 391 5 discussion discussion NOUN ejde-67 391 6 in in ADP ejde-67 391 7 this this DET ejde-67 391 8 work work NOUN ejde-67 391 9 , , PUNCT ejde-67 391 10 we we PRON ejde-67 391 11 establish establish VERB ejde-67 391 12 the the DET ejde-67 391 13 existence existence NOUN ejde-67 391 14 of of ADP ejde-67 391 15 almost almost ADV ejde-67 391 16 periodic periodic ADJ ejde-67 391 17 solutions solution NOUN ejde-67 391 18 for for ADP ejde-67 391 19 partial partial ADJ ejde-67 391 20 functional functional ADJ ejde-67 391 21 differential differential NOUN ejde-67 391 22 equations equation NOUN ejde-67 391 23 with with ADP ejde-67 391 24 stepanov stepanov VERB ejde-67 391 25 almost almost ADV ejde-67 391 26 periodic periodic ADJ ejde-67 391 27 forcing force VERB ejde-67 391 28 functions function NOUN ejde-67 391 29 . . PUNCT ejde-67 392 1 more more ADV ejde-67 392 2 specifically specifically ADV ejde-67 392 3 , , PUNCT ejde-67 392 4 we we PRON ejde-67 392 5 improve improve VERB ejde-67 392 6 the the DET ejde-67 392 7 assumptions assumption NOUN ejde-67 392 8 in in ADP ejde-67 392 9 [ [ X ejde-67 392 10 12 12 NUM ejde-67 392 11 ] ] PUNCT ejde-67 392 12 , , PUNCT ejde-67 392 13 we we PRON ejde-67 392 14 prove prove VERB ejde-67 392 15 that that SCONJ ejde-67 392 16 the the DET ejde-67 392 17 almost almost ADV ejde-67 392 18 periodicity periodicity NOUN ejde-67 392 19 of of ADP ejde-67 392 20 the the DET ejde-67 392 21 coefficients coefficient NOUN ejde-67 392 22 in in ADP ejde-67 392 23 a a DET ejde-67 392 24 weaker weak ADJ ejde-67 392 25 sense sense NOUN ejde-67 392 26 ( ( PUNCT ejde-67 392 27 stepanov stepanov VERB ejde-67 392 28 almost almost ADV ejde-67 392 29 periodicity periodicity NOUN ejde-67 392 30 ) ) PUNCT ejde-67 392 31 of of ADP ejde-67 392 32 order order NOUN ejde-67 392 33 is be AUX ejde-67 392 34 enough enough ADJ ejde-67 392 35 to to PART ejde-67 392 36 obtain obtain VERB ejde-67 392 37 solutions solution NOUN ejde-67 392 38 that that PRON ejde-67 392 39 are be AUX ejde-67 392 40 almost almost ADV ejde-67 392 41 periodic periodic ADJ ejde-67 392 42 in in ADP ejde-67 392 43 a a DET ejde-67 392 44 strong strong ADJ ejde-67 392 45 sense sense NOUN ejde-67 392 46 ( ( PUNCT ejde-67 392 47 bochner bochner NOUN ejde-67 392 48 almost almost ADV ejde-67 392 49 periodicity periodicity NOUN ejde-67 392 50 ) ) PUNCT ejde-67 392 51 . . PUNCT ejde-67 393 1 after after ADP ejde-67 393 2 that that PRON ejde-67 393 3 , , PUNCT ejde-67 393 4 we we PRON ejde-67 393 5 extend extend VERB ejde-67 393 6 our our PRON ejde-67 393 7 result result NOUN ejde-67 393 8 to to ADP ejde-67 393 9 the the DET ejde-67 393 10 almost almost ADV ejde-67 393 11 automorphic automorphic ADJ ejde-67 393 12 case case NOUN ejde-67 393 13 . . PUNCT ejde-67 394 1 we we PRON ejde-67 394 2 give give VERB ejde-67 394 3 sufficient sufficient ADJ ejde-67 394 4 conditions condition NOUN ejde-67 394 5 insuring insure VERB ejde-67 394 6 the the DET ejde-67 394 7 existence existence NOUN ejde-67 394 8 of of ADP ejde-67 394 9 almost almost ADV ejde-67 394 10 automorphic automorphic ADJ ejde-67 394 11 solutions solution NOUN ejde-67 394 12 to to ADP ejde-67 394 13 equation equation NOUN ejde-67 394 14 ( ( PUNCT ejde-67 394 15 1.3 1.3 NUM ejde-67 394 16 ) ) PUNCT ejde-67 394 17 when when SCONJ ejde-67 394 18 the the DET ejde-67 394 19 input input NOUN ejde-67 394 20 term term NOUN ejde-67 394 21 is be AUX ejde-67 394 22 only only ADV ejde-67 394 23 stepanov stepanov VERB ejde-67 394 24 almost almost ADV ejde-67 394 25 automorphic automorphic ADJ ejde-67 394 26 . . PUNCT ejde-67 395 1 to to PART ejde-67 395 2 arrive arrive VERB ejde-67 395 3 at at ADP ejde-67 395 4 our our PRON ejde-67 395 5 results result NOUN ejde-67 395 6 , , PUNCT ejde-67 395 7 we we PRON ejde-67 395 8 employ employ VERB ejde-67 395 9 the the DET ejde-67 395 10 variation variation NOUN ejde-67 395 11 of of ADP ejde-67 395 12 constant constant ADJ ejde-67 395 13 formula formula NOUN ejde-67 395 14 and and CCONJ ejde-67 395 15 fundamental fundamental ADJ ejde-67 395 16 results result NOUN ejde-67 395 17 on on ADP ejde-67 395 18 the the DET ejde-67 395 19 spectral spectral ADJ ejde-67 395 20 analysis analysis NOUN ejde-67 395 21 of of ADP ejde-67 395 22 the the DET ejde-67 395 23 solutions solution NOUN ejde-67 395 24 which which PRON ejde-67 395 25 is be AUX ejde-67 395 26 the the DET ejde-67 395 27 main main ADJ ejde-67 395 28 tool tool NOUN ejde-67 395 29 of of ADP ejde-67 395 30 this this DET ejde-67 395 31 work work NOUN ejde-67 395 32 . . PUNCT ejde-67 396 1 under under ADP ejde-67 396 2 the the DET ejde-67 396 3 hypothesis hypothesis NOUN ejde-67 396 4 that that PRON ejde-67 396 5 the the DET ejde-67 396 6 operator operator NOUN ejde-67 396 7 t t PROPN ejde-67 396 8 ( ( PUNCT ejde-67 396 9 t)l t)l NOUN ejde-67 396 10 is be AUX ejde-67 396 11 compact compact ADJ ejde-67 396 12 for for ADP ejde-67 396 13 t t PROPN ejde-67 396 14 > > X ejde-67 396 15 0 0 PROPN ejde-67 396 16 , , PUNCT ejde-67 396 17 we we PRON ejde-67 396 18 develop develop VERB ejde-67 396 19 a a DET ejde-67 396 20 new new ADJ ejde-67 396 21 fundamental fundamental ADJ ejde-67 396 22 reduction reduction NOUN ejde-67 396 23 principle principle NOUN ejde-67 396 24 that that PRON ejde-67 396 25 is be AUX ejde-67 396 26 different different ADJ ejde-67 396 27 from from ADP ejde-67 396 28 the the DET ejde-67 396 29 one one NUM ejde-67 396 30 in in ADP ejde-67 396 31 [ [ X ejde-67 396 32 11 11 NUM ejde-67 396 33 ] ] PUNCT ejde-67 396 34 . . PUNCT ejde-67 397 1 indeed indeed ADV ejde-67 397 2 , , PUNCT ejde-67 397 3 to to PART ejde-67 397 4 establish establish VERB ejde-67 397 5 the the DET ejde-67 397 6 reduction reduction NOUN ejde-67 397 7 principle principle NOUN ejde-67 397 8 we we PRON ejde-67 397 9 take take VERB ejde-67 397 10 an an DET ejde-67 397 11 approach approach NOUN ejde-67 397 12 similar similar ADJ ejde-67 397 13 to to ADP ejde-67 397 14 that that PRON ejde-67 397 15 in in ADP ejde-67 397 16 [ [ X ejde-67 397 17 12 12 NUM ejde-67 397 18 ] ] PUNCT ejde-67 397 19 and and CCONJ ejde-67 397 20 [ [ X ejde-67 397 21 11 11 NUM ejde-67 397 22 ] ] PUNCT ejde-67 397 23 without without ADP ejde-67 397 24 using use VERB ejde-67 397 25 the the DET ejde-67 397 26 compactness compactness NOUN ejde-67 397 27 of of ADP ejde-67 397 28 c0 c0 PROPN ejde-67 397 29 - - PUNCT ejde-67 397 30 semigroup semigroup PROPN ejde-67 397 31 ( ( PUNCT ejde-67 397 32 t t PROPN ejde-67 397 33 ( ( PUNCT ejde-67 397 34 t))t≥0 t))t≥0 PROPN ejde-67 397 35 . . PUNCT ejde-67 398 1 moreover moreover ADV ejde-67 398 2 , , PUNCT ejde-67 398 3 we we PRON ejde-67 398 4 prove prove VERB ejde-67 398 5 the the DET ejde-67 398 6 fundamental fundamental ADJ ejde-67 398 7 theorem theorem NOUN ejde-67 398 8 of of ADP ejde-67 398 9 existence existence NOUN ejde-67 398 10 of of ADP ejde-67 398 11 almost almost ADV ejde-67 398 12 periodic periodic ADJ ejde-67 398 13 solutions solution NOUN ejde-67 398 14 and and CCONJ ejde-67 398 15 almost almost ADV ejde-67 398 16 automorphic automorphic ADJ ejde-67 398 17 solutions solution NOUN ejde-67 398 18 . . PUNCT ejde-67 399 1 at at ADP ejde-67 399 2 the the DET ejde-67 399 3 end end NOUN ejde-67 399 4 , , PUNCT ejde-67 399 5 we we PRON ejde-67 399 6 illustrate illustrate VERB ejde-67 399 7 our our PRON ejde-67 399 8 theoretical theoretical ADJ ejde-67 399 9 result result NOUN ejde-67 399 10 to to ADP ejde-67 399 11 a a DET ejde-67 399 12 transportation transportation NOUN ejde-67 399 13 equation equation NOUN ejde-67 399 14 . . PUNCT ejde-67 400 1 acknowledgments acknowledgment NOUN ejde-67 400 2 . . PUNCT ejde-67 401 1 the the DET ejde-67 401 2 authors author NOUN ejde-67 401 3 want want VERB ejde-67 401 4 to to PART ejde-67 401 5 thank thank VERB ejde-67 401 6 the the DET ejde-67 401 7 anonymous anonymous ADJ ejde-67 401 8 referees referee NOUN ejde-67 401 9 for for ADP ejde-67 401 10 carefully carefully ADV ejde-67 401 11 reading read VERB ejde-67 401 12 the the DET ejde-67 401 13 work work NOUN ejde-67 401 14 and and CCONJ ejde-67 401 15 for for ADP ejde-67 401 16 providing provide VERB ejde-67 401 17 insightful insightful ADJ ejde-67 401 18 feedback feedback NOUN ejde-67 401 19 . . PUNCT ejde-67 402 1 16 16 NUM ejde-67 402 2 m. m. NOUN ejde-67 402 3 el el PROPN ejde-67 402 4 attaouy attaouy PROPN ejde-67 402 5 , , PUNCT ejde-67 402 6 k. k. PROPN ejde-67 402 7 ezzinbi ezzinbi PROPN ejde-67 402 8 , , PUNCT ejde-67 402 9 g. g. PROPN ejde-67 402 10 m. m. PROPN ejde-67 402 11 n’guérékata n’guérékata PROPN ejde-67 402 12 ejde-2023/39 ejde-2023/39 PROPN ejde-67 402 13 references reference NOUN ejde-67 402 14 [ [ X ejde-67 402 15 1 1 NUM ejde-67 402 16 ] ] X ejde-67 402 17 e. e. PROPN ejde-67 402 18 ait ait PROPN ejde-67 402 19 dads dad VERB ejde-67 402 20 , , PUNCT ejde-67 402 21 b. b. PROPN ejde-67 402 22 es es PROPN ejde-67 402 23 - - PUNCT ejde-67 402 24 sebbar sebbar NOUN ejde-67 402 25 , , PUNCT ejde-67 402 26 k. k. PROPN ejde-67 402 27 ezzinbi ezzinbi PROPN ejde-67 402 28 , , PUNCT ejde-67 402 29 m. m. NOUN ejde-67 402 30 ziat ziat PROPN ejde-67 402 31 ; ; PUNCT ejde-67 402 32 behavior behavior NOUN ejde-67 402 33 of of ADP ejde-67 402 34 bounded bounded ADJ ejde-67 402 35 solutions solution NOUN ejde-67 402 36 for for ADP ejde-67 402 37 some some DET ejde-67 402 38 almost almost ADV ejde-67 402 39 periodic periodic ADJ ejde-67 402 40 neutral neutral ADJ ejde-67 402 41 partial partial ADJ ejde-67 402 42 functional functional ADJ ejde-67 402 43 differential differential NOUN ejde-67 402 44 equations equation NOUN ejde-67 402 45 , , PUNCT ejde-67 402 46 mathematical mathematical ADJ ejde-67 402 47 methods method NOUN ejde-67 402 48 in in ADP ejde-67 402 49 the the DET ejde-67 402 50 applied apply VERB ejde-67 402 51 sciences science NOUN ejde-67 402 52 , , PUNCT ejde-67 402 53 40 40 NUM ejde-67 402 54 ( ( PUNCT ejde-67 402 55 7 7 NUM ejde-67 402 56 ) ) PUNCT ejde-67 402 57 ( ( PUNCT ejde-67 402 58 2017 2017 NUM ejde-67 402 59 ) ) PUNCT ejde-67 402 60 , , PUNCT ejde-67 402 61 2377 2377 NUM ejde-67 402 62 - - SYM ejde-67 402 63 2397 2397 NUM ejde-67 402 64 . . PUNCT ejde-67 403 1 [ [ X ejde-67 403 2 2 2 NUM ejde-67 403 3 ] ] X ejde-67 403 4 r. r. PROPN ejde-67 403 5 benkhalti benkhalti PROPN ejde-67 403 6 , , PUNCT ejde-67 403 7 b. b. PROPN ejde-67 403 8 es es PROPN ejde-67 403 9 - - PUNCT ejde-67 403 10 sebbar sebbar NOUN ejde-67 403 11 , , PUNCT ejde-67 403 12 k. k. PROPN ejde-67 403 13 ezzinbi ezzinbi NOUN ejde-67 403 14 ; ; PUNCT ejde-67 403 15 on on ADP ejde-67 403 16 a a DET ejde-67 403 17 bohr bohr NOUN ejde-67 403 18 - - PUNCT ejde-67 403 19 neugebauer neugebauer NOUN ejde-67 403 20 property property NOUN ejde-67 403 21 for for ADP ejde-67 403 22 some some DET ejde-67 403 23 almost almost ADV ejde-67 403 24 automorphic automorphic ADJ ejde-67 403 25 abstract abstract ADJ ejde-67 403 26 delay delay NOUN ejde-67 403 27 equations equation NOUN ejde-67 403 28 , , PUNCT ejde-67 403 29 journal journal NOUN ejde-67 403 30 of of ADP ejde-67 403 31 integral integral ADJ ejde-67 403 32 equations equation NOUN ejde-67 403 33 and and CCONJ ejde-67 403 34 applications application NOUN ejde-67 403 35 , , PUNCT ejde-67 403 36 30 30 NUM ejde-67 403 37 ( ( PUNCT ejde-67 403 38 3 3 NUM ejde-67 403 39 ) ) PUNCT ejde-67 403 40 ( ( PUNCT ejde-67 403 41 2018 2018 NUM ejde-67 403 42 ) ) PUNCT ejde-67 403 43 , , PUNCT ejde-67 403 44 313 313 NUM ejde-67 403 45 - - SYM ejde-67 403 46 345 345 NUM ejde-67 403 47 . . PUNCT ejde-67 404 1 [ [ X ejde-67 404 2 3 3 X ejde-67 404 3 ] ] X ejde-67 404 4 h. h. PROPN ejde-67 404 5 bohr bohr PROPN ejde-67 404 6 ; ; PUNCT ejde-67 404 7 zur zur NOUN ejde-67 404 8 theorie theorie PROPN ejde-67 404 9 der der PROPN ejde-67 404 10 fastperiodischen fastperiodischen PROPN ejde-67 404 11 funktionen funktionen PROPN ejde-67 404 12 , , PUNCT ejde-67 404 13 acta acta PROPN ejde-67 404 14 mathematica mathematica PROPN ejde-67 404 15 , , PUNCT ejde-67 404 16 46 46 NUM ejde-67 404 17 ( ( PUNCT ejde-67 404 18 1 1 NUM ejde-67 404 19 - - SYM ejde-67 404 20 2 2 NUM ejde-67 404 21 ) ) PUNCT ejde-67 404 22 ( ( PUNCT ejde-67 404 23 1925 1925 NUM ejde-67 404 24 ) ) PUNCT ejde-67 404 25 , , PUNCT ejde-67 404 26 101 101 NUM ejde-67 404 27 - - SYM ejde-67 404 28 214 214 NUM ejde-67 404 29 . . PUNCT ejde-67 405 1 [ [ X ejde-67 405 2 4 4 X ejde-67 405 3 ] ] X ejde-67 405 4 s. s. PROPN ejde-67 405 5 bochner bochner PROPN ejde-67 405 6 ; ; PUNCT ejde-67 405 7 continuous continuous ADJ ejde-67 405 8 mappings mapping NOUN ejde-67 405 9 of of ADP ejde-67 405 10 almost almost ADV ejde-67 405 11 automorphic automorphic ADJ ejde-67 405 12 and and CCONJ ejde-67 405 13 almost almost ADV ejde-67 405 14 periodic periodic ADJ ejde-67 405 15 functions function NOUN ejde-67 405 16 , , PUNCT ejde-67 405 17 proceedings proceeding NOUN ejde-67 405 18 of of ADP ejde-67 405 19 the the DET ejde-67 405 20 national national PROPN ejde-67 405 21 academy academy PROPN ejde-67 405 22 of of ADP ejde-67 405 23 sciences sciences PROPN ejde-67 405 24 of of ADP ejde-67 405 25 the the DET ejde-67 405 26 united united PROPN ejde-67 405 27 states states PROPN ejde-67 405 28 of of ADP ejde-67 405 29 america america PROPN ejde-67 405 30 , , PUNCT ejde-67 405 31 52 52 NUM ejde-67 405 32 , , PUNCT ejde-67 405 33 907910 907910 NUM ejde-67 405 34 , , PUNCT ejde-67 405 35 1964 1964 NUM ejde-67 405 36 . . PUNCT ejde-67 406 1 [ [ X ejde-67 406 2 5 5 NUM ejde-67 406 3 ] ] X ejde-67 406 4 n. n. PROPN ejde-67 406 5 boukli boukli PROPN ejde-67 406 6 , , PUNCT ejde-67 406 7 k. k. PROPN ejde-67 406 8 ezzinbi ezzinbi NOUN ejde-67 406 9 ; ; PUNCT ejde-67 406 10 weighted weight VERB ejde-67 406 11 pseudo pseudo NOUN ejde-67 406 12 almost almost ADV ejde-67 406 13 periodic periodic ADJ ejde-67 406 14 solutions solution NOUN ejde-67 406 15 for for ADP ejde-67 406 16 some some DET ejde-67 406 17 partial partial ADJ ejde-67 406 18 functional functional ADJ ejde-67 406 19 differential differential NOUN ejde-67 406 20 equations equation NOUN ejde-67 406 21 , , PUNCT ejde-67 406 22 nonlinear nonlinear ADJ ejde-67 406 23 analysis analysis NOUN ejde-67 406 24 : : PUNCT ejde-67 406 25 theory theory NOUN ejde-67 406 26 , , PUNCT ejde-67 406 27 methods method NOUN ejde-67 406 28 and and CCONJ ejde-67 406 29 applications application NOUN ejde-67 406 30 , , PUNCT ejde-67 406 31 71 71 NUM ejde-67 406 32 ( ( PUNCT ejde-67 406 33 2009 2009 NUM ejde-67 406 34 ) ) PUNCT ejde-67 406 35 , , PUNCT ejde-67 406 36 3612 3612 NUM ejde-67 406 37 - - SYM ejde-67 406 38 3621 3621 NUM ejde-67 406 39 . . PUNCT ejde-67 407 1 [ [ X ejde-67 407 2 6 6 NUM ejde-67 407 3 ] ] PUNCT ejde-67 407 4 p. p. NOUN ejde-67 407 5 cieutat cieutat PROPN ejde-67 407 6 , , PUNCT ejde-67 407 7 k. k. PROPN ejde-67 407 8 ezzinbi ezzinbi NOUN ejde-67 407 9 ; ; PUNCT ejde-67 407 10 almost almost ADV ejde-67 407 11 automorphic automorphic ADJ ejde-67 407 12 solutions solution NOUN ejde-67 407 13 for for ADP ejde-67 407 14 some some DET ejde-67 407 15 evolution evolution NOUN ejde-67 407 16 equations equation NOUN ejde-67 407 17 through through ADP ejde-67 407 18 the the DET ejde-67 407 19 minimizing minimizing NOUN ejde-67 407 20 for for ADP ejde-67 407 21 some some DET ejde-67 407 22 subvariant subvariant ADJ ejde-67 407 23 functional functional ADJ ejde-67 407 24 , , PUNCT ejde-67 407 25 applications application NOUN ejde-67 407 26 to to PART ejde-67 407 27 heat heat VERB ejde-67 407 28 and and CCONJ ejde-67 407 29 wave wave NOUN ejde-67 407 30 equations equation NOUN ejde-67 407 31 with with ADP ejde-67 407 32 nonlinearities nonlinearitie NOUN ejde-67 407 33 , , PUNCT ejde-67 407 34 journal journal NOUN ejde-67 407 35 of of ADP ejde-67 407 36 functional functional ADJ ejde-67 407 37 analysis analysis NOUN ejde-67 407 38 260(9 260(9 NUM ejde-67 407 39 ) ) PUNCT ejde-67 407 40 , , PUNCT ejde-67 407 41 2598 2598 NUM ejde-67 407 42 - - SYM ejde-67 407 43 2634 2634 NUM ejde-67 407 44 , , PUNCT ejde-67 407 45 ( ( PUNCT ejde-67 407 46 2011 2011 NUM ejde-67 407 47 ) ) PUNCT ejde-67 407 48 . . PUNCT ejde-67 408 1 [ [ X ejde-67 408 2 7 7 X ejde-67 408 3 ] ] X ejde-67 408 4 c. c. NOUN ejde-67 408 5 corduneanu corduneanu PROPN ejde-67 408 6 ; ; PUNCT ejde-67 408 7 principles principle NOUN ejde-67 408 8 of of ADP ejde-67 408 9 differential differential ADJ ejde-67 408 10 and and CCONJ ejde-67 408 11 integral integral ADJ ejde-67 408 12 equations equation NOUN ejde-67 408 13 , , PUNCT ejde-67 408 14 american american PROPN ejde-67 408 15 mathematical mathematical ADJ ejde-67 408 16 society society NOUN ejde-67 408 17 , , PUNCT ejde-67 408 18 providence providence NOUN ejde-67 408 19 , , PUNCT ejde-67 408 20 2008 2008 NUM ejde-67 408 21 . . PUNCT ejde-67 409 1 [ [ X ejde-67 409 2 8 8 X ejde-67 409 3 ] ] X ejde-67 409 4 t. t. PROPN ejde-67 409 5 diagana diagana PROPN ejde-67 409 6 ; ; PUNCT ejde-67 409 7 stepanov stepanov VERB ejde-67 409 8 - - PUNCT ejde-67 409 9 like like ADJ ejde-67 409 10 pseudo pseudo NOUN ejde-67 409 11 - - ADJ ejde-67 409 12 almost almost ADV ejde-67 409 13 periodicity periodicity NOUN ejde-67 409 14 and and CCONJ ejde-67 409 15 its its PRON ejde-67 409 16 applications application NOUN ejde-67 409 17 to to ADP ejde-67 409 18 some some DET ejde-67 409 19 nonautonomous nonautonomous ADJ ejde-67 409 20 differential differential NOUN ejde-67 409 21 equations equation NOUN ejde-67 409 22 , , PUNCT ejde-67 409 23 nonlinear nonlinear ADJ ejde-67 409 24 analysis analysis NOUN ejde-67 409 25 : : PUNCT ejde-67 409 26 theory theory NOUN ejde-67 409 27 , , PUNCT ejde-67 409 28 methods method NOUN ejde-67 409 29 and and CCONJ ejde-67 409 30 applications application NOUN ejde-67 409 31 ; ; PUNCT ejde-67 409 32 69 69 NUM ejde-67 409 33 ( ( PUNCT ejde-67 409 34 12 12 NUM ejde-67 409 35 ) ) PUNCT ejde-67 409 36 ( ( PUNCT ejde-67 409 37 2008 2008 NUM ejde-67 409 38 ) ) PUNCT ejde-67 409 39 , , PUNCT ejde-67 409 40 4277 4277 NUM ejde-67 409 41 - - SYM ejde-67 409 42 428 428 NUM ejde-67 409 43 . . PUNCT ejde-67 410 1 [ [ X ejde-67 410 2 9 9 NUM ejde-67 410 3 ] ] PUNCT ejde-67 410 4 t. t. PROPN ejde-67 410 5 diagana diagana PROPN ejde-67 410 6 , , PUNCT ejde-67 410 7 g. g. PROPN ejde-67 410 8 m. m. PROPN ejde-67 410 9 n’guérékata n’guérékata PROPN ejde-67 410 10 ; ; PUNCT ejde-67 410 11 stepanov stepanov VERB ejde-67 410 12 - - PUNCT ejde-67 410 13 like like ADJ ejde-67 410 14 almost almost ADV ejde-67 410 15 automorphic automorphic ADJ ejde-67 410 16 functions function NOUN ejde-67 410 17 and and CCONJ ejde-67 410 18 applications application NOUN ejde-67 410 19 to to ADP ejde-67 410 20 some some DET ejde-67 410 21 semilinear semilinear NOUN ejde-67 410 22 equations equation NOUN ejde-67 410 23 , , PUNCT ejde-67 410 24 applicable applicable ADJ ejde-67 410 25 analysis analysis NOUN ejde-67 410 26 , , PUNCT ejde-67 410 27 86 86 NUM ejde-67 410 28 ( ( PUNCT ejde-67 410 29 2007 2007 NUM ejde-67 410 30 ) ) PUNCT ejde-67 410 31 , , PUNCT ejde-67 410 32 723 723 NUM ejde-67 410 33 - - SYM ejde-67 410 34 733 733 NUM ejde-67 410 35 . . PUNCT ejde-67 411 1 [ [ X ejde-67 411 2 10 10 NUM ejde-67 411 3 ] ] PUNCT ejde-67 411 4 k. k. PROPN ejde-67 411 5 j. j. PROPN ejde-67 411 6 engel engel PROPN ejde-67 411 7 , , PUNCT ejde-67 411 8 r. r. PROPN ejde-67 411 9 nagel nagel PROPN ejde-67 411 10 ; ; PUNCT ejde-67 411 11 one one NUM ejde-67 411 12 - - PUNCT ejde-67 411 13 parameter parameter NOUN ejde-67 411 14 semigroups semigroup NOUN ejde-67 411 15 for for ADP ejde-67 411 16 linear linear PROPN ejde-67 411 17 evolution evolution NOUN ejde-67 411 18 equations equation NOUN ejde-67 411 19 , , PUNCT ejde-67 411 20 springer springer NOUN ejde-67 411 21 , , PUNCT ejde-67 411 22 new new PROPN ejde-67 411 23 york york PROPN ejde-67 411 24 , , PUNCT ejde-67 411 25 2000 2000 NUM ejde-67 411 26 . . PUNCT ejde-67 412 1 [ [ X ejde-67 412 2 11 11 NUM ejde-67 412 3 ] ] PUNCT ejde-67 412 4 k. k. PROPN ejde-67 412 5 ezzinbi ezzinbi PROPN ejde-67 412 6 , , PUNCT ejde-67 412 7 g. g. PROPN ejde-67 412 8 m. m. PROPN ejde-67 412 9 n’guérékata n’guérékata PROPN ejde-67 412 10 ; ; PUNCT ejde-67 412 11 almost almost ADV ejde-67 412 12 automorphic automorphic ADJ ejde-67 412 13 solutions solution NOUN ejde-67 412 14 for for ADP ejde-67 412 15 some some DET ejde-67 412 16 partial partial ADJ ejde-67 412 17 functional functional ADJ ejde-67 412 18 differential differential NOUN ejde-67 412 19 equations equation NOUN ejde-67 412 20 , , PUNCT ejde-67 412 21 journal journal NOUN ejde-67 412 22 of of ADP ejde-67 412 23 mathematical mathematical ADJ ejde-67 412 24 analysis analysis NOUN ejde-67 412 25 and and CCONJ ejde-67 412 26 applications application NOUN ejde-67 412 27 , , PUNCT ejde-67 412 28 328 328 NUM ejde-67 412 29 ( ( PUNCT ejde-67 412 30 2007 2007 NUM ejde-67 412 31 ) ) PUNCT ejde-67 412 32 , , PUNCT ejde-67 412 33 344358 344358 NUM ejde-67 412 34 . . PUNCT ejde-67 413 1 [ [ X ejde-67 413 2 12 12 NUM ejde-67 413 3 ] ] X ejde-67 413 4 h. h. PROPN ejde-67 413 5 henriquez henriquez PROPN ejde-67 413 6 , , PUNCT ejde-67 413 7 c. c. PROPN ejde-67 413 8 cuevas cuevas PROPN ejde-67 413 9 , , PUNCT ejde-67 413 10 a. a. NOUN ejde-67 413 11 caicedo caicedo PROPN ejde-67 413 12 ; ; PUNCT ejde-67 413 13 almost almost ADV ejde-67 413 14 periodic periodic ADJ ejde-67 413 15 solutions solution NOUN ejde-67 413 16 of of ADP ejde-67 413 17 partial partial ADJ ejde-67 413 18 differential differential ADJ ejde-67 413 19 equations equation NOUN ejde-67 413 20 with with ADP ejde-67 413 21 delay delay NOUN ejde-67 413 22 , , PUNCT ejde-67 413 23 advances advance NOUN ejde-67 413 24 in in ADP ejde-67 413 25 difference difference NOUN ejde-67 413 26 equations equation NOUN ejde-67 413 27 , , PUNCT ejde-67 413 28 2015(1 2015(1 NUM ejde-67 413 29 ) ) PUNCT ejde-67 413 30 , , PUNCT ejde-67 413 31 1 1 NUM ejde-67 413 32 - - SYM ejde-67 413 33 15 15 NUM ejde-67 413 34 , , PUNCT ejde-67 413 35 ( ( PUNCT ejde-67 413 36 2015 2015 NUM ejde-67 413 37 ) ) PUNCT ejde-67 413 38 . . PUNCT ejde-67 414 1 [ [ X ejde-67 414 2 13 13 NUM ejde-67 414 3 ] ] X ejde-67 414 4 y. y. NOUN ejde-67 414 5 hino hino PROPN ejde-67 414 6 , , PUNCT ejde-67 414 7 t. t. PROPN ejde-67 414 8 naito naito PROPN ejde-67 414 9 , , PUNCT ejde-67 414 10 n. n. PROPN ejde-67 414 11 van van PROPN ejde-67 414 12 minh minh PROPN ejde-67 414 13 , , PUNCT ejde-67 414 14 j. j. PROPN ejde-67 414 15 s. s. PROPN ejde-67 414 16 shin shin PROPN ejde-67 414 17 ; ; PUNCT ejde-67 414 18 almost almost ADV ejde-67 414 19 periodic periodic ADJ ejde-67 414 20 solutions solution NOUN ejde-67 414 21 of of ADP ejde-67 414 22 differential differential ADJ ejde-67 414 23 equations equation NOUN ejde-67 414 24 in in ADP ejde-67 414 25 banach banach NOUN ejde-67 414 26 spaces space NOUN ejde-67 414 27 , , PUNCT ejde-67 414 28 taylor taylor PROPN ejde-67 414 29 & & CCONJ ejde-67 414 30 francis francis PROPN ejde-67 414 31 , , PUNCT ejde-67 414 32 london london PROPN ejde-67 414 33 , , PUNCT ejde-67 414 34 2002 2002 NUM ejde-67 414 35 . . PUNCT ejde-67 415 1 [ [ X ejde-67 415 2 14 14 NUM ejde-67 415 3 ] ] PUNCT ejde-67 415 4 j. j. PROPN ejde-67 415 5 liu liu PROPN ejde-67 415 6 , , PUNCT ejde-67 415 7 g. g. PROPN ejde-67 415 8 m. m. PROPN ejde-67 415 9 n’guérékata n’guérékata PROPN ejde-67 415 10 , , PUNCT ejde-67 415 11 nguyen nguyen PROPN ejde-67 415 12 van van PROPN ejde-67 415 13 minh minh PROPN ejde-67 415 14 ; ; PUNCT ejde-67 415 15 a a DET ejde-67 415 16 massera massera NOUN ejde-67 415 17 type type NOUN ejde-67 415 18 theorem theorem NOUN ejde-67 415 19 for for ADP ejde-67 415 20 almost almost ADV ejde-67 415 21 automorphic automorphic ADJ ejde-67 415 22 solutions solution NOUN ejde-67 415 23 of of ADP ejde-67 415 24 differential differential ADJ ejde-67 415 25 equations equation NOUN ejde-67 415 26 , , PUNCT ejde-67 415 27 journal journal NOUN ejde-67 415 28 of of ADP ejde-67 415 29 mathematical mathematical ADJ ejde-67 415 30 analysis analysis NOUN ejde-67 415 31 and and CCONJ ejde-67 415 32 applications application NOUN ejde-67 415 33 , , PUNCT ejde-67 415 34 299 299 NUM ejde-67 415 35 ( ( PUNCT ejde-67 415 36 2004 2004 NUM ejde-67 415 37 ) ) PUNCT ejde-67 415 38 , , PUNCT ejde-67 415 39 587 587 NUM ejde-67 415 40 - - SYM ejde-67 415 41 599 599 NUM ejde-67 415 42 . . PUNCT ejde-67 416 1 [ [ X ejde-67 416 2 15 15 NUM ejde-67 416 3 ] ] X ejde-67 416 4 g. g. PROPN ejde-67 416 5 m. m. PROPN ejde-67 416 6 n’guérékata n’guérékata PROPN ejde-67 416 7 ; ; PUNCT ejde-67 416 8 almost almost ADV ejde-67 416 9 periodic periodic ADJ ejde-67 416 10 and and CCONJ ejde-67 416 11 almost almost ADV ejde-67 416 12 automorphic automorphic ADJ ejde-67 416 13 functions function NOUN ejde-67 416 14 in in ADP ejde-67 416 15 abstract abstract ADJ ejde-67 416 16 spaces space NOUN ejde-67 416 17 , , PUNCT ejde-67 416 18 springer springer NOUN ejde-67 416 19 second second PROPN ejde-67 416 20 edition edition PROPN ejde-67 416 21 new new PROPN ejde-67 416 22 york york PROPN ejde-67 416 23 , , PUNCT ejde-67 416 24 2021 2021 NUM ejde-67 416 25 . . PUNCT ejde-67 417 1 [ [ X ejde-67 417 2 16 16 NUM ejde-67 417 3 ] ] X ejde-67 417 4 g. g. PROPN ejde-67 417 5 m. m. PROPN ejde-67 417 6 n’guérékata n’guérékata PROPN ejde-67 417 7 ; ; PUNCT ejde-67 417 8 existence existence NOUN ejde-67 417 9 and and CCONJ ejde-67 417 10 uniqueness uniqueness NOUN ejde-67 417 11 of of ADP ejde-67 417 12 almost almost ADV ejde-67 417 13 automorphic automorphic ADJ ejde-67 417 14 mild mild ADJ ejde-67 417 15 solutions solution NOUN ejde-67 417 16 to to ADP ejde-67 417 17 some some DET ejde-67 417 18 semilinear semilinear ADJ ejde-67 417 19 abstract abstract ADJ ejde-67 417 20 differential differential ADJ ejde-67 417 21 equations equation NOUN ejde-67 417 22 , , PUNCT ejde-67 417 23 semigroup semigroup PROPN ejde-67 417 24 forum forum PROPN ejde-67 417 25 , , PUNCT ejde-67 417 26 69 69 NUM ejde-67 417 27 ( ( PUNCT ejde-67 417 28 2004 2004 NUM ejde-67 417 29 ) ) PUNCT ejde-67 417 30 , , PUNCT ejde-67 417 31 80 80 NUM ejde-67 417 32 - - SYM ejde-67 417 33 86 86 NUM ejde-67 417 34 . . PUNCT ejde-67 418 1 [ [ X ejde-67 418 2 17 17 NUM ejde-67 418 3 ] ] X ejde-67 418 4 g. g. PROPN ejde-67 418 5 m. m. PROPN ejde-67 418 6 n’guérékata n’guérékata PROPN ejde-67 418 7 , , PUNCT ejde-67 418 8 a. a. NOUN ejde-67 418 9 pankov pankov NOUN ejde-67 418 10 ; ; PUNCT ejde-67 418 11 stepanov stepanov VERB ejde-67 418 12 - - PUNCT ejde-67 418 13 like like ADJ ejde-67 418 14 almost almost ADV ejde-67 418 15 automorphic automorphic ADJ ejde-67 418 16 functions function NOUN ejde-67 418 17 and and CCONJ ejde-67 418 18 monotone monotone ADJ ejde-67 418 19 evolution evolution NOUN ejde-67 418 20 equations equation NOUN ejde-67 418 21 , , PUNCT ejde-67 418 22 nonlinear nonlinear ADJ ejde-67 418 23 analysis analysis NOUN ejde-67 418 24 : : PUNCT ejde-67 418 25 theory theory NOUN ejde-67 418 26 , , PUNCT ejde-67 418 27 methods method NOUN ejde-67 418 28 and and CCONJ ejde-67 418 29 applications application NOUN ejde-67 418 30 , , PUNCT ejde-67 418 31 68 68 NUM ejde-67 418 32 ( ( PUNCT ejde-67 418 33 2008 2008 NUM ejde-67 418 34 ) ) PUNCT ejde-67 418 35 , , PUNCT ejde-67 418 36 2658 2658 NUM ejde-67 418 37 – – PUNCT ejde-67 418 38 2667 2667 NUM ejde-67 418 39 . . PUNCT ejde-67 419 1 [ [ X ejde-67 419 2 18 18 NUM ejde-67 419 3 ] ] PUNCT ejde-67 419 4 m. m. NOUN ejde-67 419 5 tarallo tarallo PROPN ejde-67 419 6 ; ; PUNCT ejde-67 419 7 a a DET ejde-67 419 8 stepanov stepanov ADJ ejde-67 419 9 version version NOUN ejde-67 419 10 for for ADP ejde-67 419 11 favard favard PROPN ejde-67 419 12 theory theory NOUN ejde-67 419 13 , , PUNCT ejde-67 419 14 archiv archiv PROPN ejde-67 419 15 der der PROPN ejde-67 419 16 mathematik mathematik PROPN ejde-67 419 17 , , PUNCT ejde-67 419 18 90 90 NUM ejde-67 419 19 , , PUNCT ejde-67 419 20 53 53 NUM ejde-67 419 21 - - SYM ejde-67 419 22 59 59 NUM ejde-67 419 23 , , PUNCT ejde-67 419 24 2008 2008 NUM ejde-67 419 25 . . PUNCT ejde-67 420 1 [ [ X ejde-67 420 2 19 19 NUM ejde-67 420 3 ] ] X ejde-67 420 4 j. j. PROPN ejde-67 420 5 wu wu PROPN ejde-67 420 6 ; ; PUNCT ejde-67 420 7 theory theory NOUN ejde-67 420 8 and and CCONJ ejde-67 420 9 applications application NOUN ejde-67 420 10 of of ADP ejde-67 420 11 partial partial ADJ ejde-67 420 12 functional functional ADJ ejde-67 420 13 differential differential NOUN ejde-67 420 14 equations equation NOUN ejde-67 420 15 , , PUNCT ejde-67 420 16 springer springer NOUN ejde-67 420 17 , , PUNCT ejde-67 420 18 new new PROPN ejde-67 420 19 york york PROPN ejde-67 420 20 , , PUNCT ejde-67 420 21 1996 1996 NUM ejde-67 420 22 . . PUNCT ejde-67 421 1 [ [ X ejde-67 421 2 20 20 NUM ejde-67 421 3 ] ] PUNCT ejde-67 421 4 l. l. PROPN ejde-67 421 5 zhang zhang PROPN ejde-67 421 6 , , PUNCT ejde-67 421 7 y. y. PROPN ejde-67 421 8 xu xu PROPN ejde-67 421 9 ; ; PUNCT ejde-67 421 10 weighted weight VERB ejde-67 421 11 pseudo pseudo NOUN ejde-67 421 12 almost almost ADV ejde-67 421 13 periodic periodic ADJ ejde-67 421 14 solutions solution NOUN ejde-67 421 15 for for ADP ejde-67 421 16 functional functional ADJ ejde-67 421 17 differential differential ADJ ejde-67 421 18 equations equation NOUN ejde-67 421 19 , , PUNCT ejde-67 421 20 electronic electronic ADJ ejde-67 421 21 journal journal NOUN ejde-67 421 22 of of ADP ejde-67 421 23 differential differential ADJ ejde-67 421 24 equation equation NOUN ejde-67 421 25 , , PUNCT ejde-67 421 26 2007 2007 NUM ejde-67 421 27 ( ( PUNCT ejde-67 421 28 2007 2007 NUM ejde-67 421 29 ) ) PUNCT ejde-67 422 1 no no INTJ ejde-67 422 2 . . NOUN ejde-67 423 1 146 146 NUM ejde-67 423 2 , , PUNCT ejde-67 423 3 1 1 NUM ejde-67 423 4 - - SYM ejde-67 423 5 7 7 NUM ejde-67 423 6 . . PUNCT ejde-67 424 1 meryem meryem PROPN ejde-67 424 2 el el PROPN ejde-67 424 3 attaouy attaouy VERB ejde-67 424 4 cadi cadi PROPN ejde-67 424 5 ayyad ayyad PROPN ejde-67 424 6 university university PROPN ejde-67 424 7 , , PUNCT ejde-67 424 8 faculty faculty NOUN ejde-67 424 9 of of ADP ejde-67 424 10 science science NOUN ejde-67 424 11 semlalia semlalia PROPN ejde-67 424 12 , , PUNCT ejde-67 424 13 department department NOUN ejde-67 424 14 of of ADP ejde-67 424 15 mathematics mathematics PROPN ejde-67 424 16 , , PUNCT ejde-67 424 17 bp bp PROPN ejde-67 424 18 2390 2390 NUM ejde-67 424 19 marrakech marrakech X ejde-67 424 20 , , PUNCT ejde-67 424 21 morocco morocco PROPN ejde-67 424 22 email email NOUN ejde-67 424 23 address address NOUN ejde-67 424 24 : : PUNCT ejde-67 424 25 meryemelattaouy@gmail.com meryemelattaouy@gmail.com X ejde-67 425 1 ejde-2023/39 ejde-2023/39 ADJ ejde-67 425 2 reduction reduction NOUN ejde-67 425 3 principle principle NOUN ejde-67 425 4 17 17 NUM ejde-67 425 5 khalil khalil PROPN ejde-67 425 6 ezzinbi ezzinbi PROPN ejde-67 425 7 cadi cadi PROPN ejde-67 425 8 ayyad ayyad PROPN ejde-67 425 9 university university PROPN ejde-67 425 10 , , PUNCT ejde-67 425 11 faculty faculty NOUN ejde-67 425 12 of of ADP ejde-67 425 13 science science NOUN ejde-67 425 14 semlalia semlalia PROPN ejde-67 425 15 , , PUNCT ejde-67 425 16 department department NOUN ejde-67 425 17 of of ADP ejde-67 425 18 mathematics mathematics PROPN ejde-67 425 19 , , PUNCT ejde-67 425 20 bp bp PROPN ejde-67 425 21 2390 2390 NUM ejde-67 425 22 marrakech marrakech PROPN ejde-67 425 23 , , PUNCT ejde-67 425 24 morocco morocco PROPN ejde-67 425 25 . . PUNCT ejde-67 426 1 umi umi PROPN ejde-67 426 2 209 209 NUM ejde-67 426 3 ummisco ummisco ADJ ejde-67 426 4 , , PUNCT ejde-67 426 5 computer computer NOUN ejde-67 426 6 and and CCONJ ejde-67 426 7 mathematical mathematical ADJ ejde-67 426 8 modeling modeling NOUN ejde-67 426 9 of of ADP ejde-67 426 10 complex complex ADJ ejde-67 426 11 systems system NOUN ejde-67 426 12 , , PUNCT ejde-67 426 13 irdbondy irdbondy ADJ ejde-67 426 14 paris paris PROPN ejde-67 426 15 email email NOUN ejde-67 426 16 address address NOUN ejde-67 426 17 : : PUNCT ejde-67 426 18 ezzinbi@uca.ac.ma ezzinbi@uca.ac.ma PROPN ejde-67 426 19 gaston gaston PROPN ejde-67 426 20 mandata mandata PROPN ejde-67 426 21 n’guérékata n’guérékata PROPN ejde-67 426 22 neerlab neerlab PROPN ejde-67 426 23 , , PUNCT ejde-67 426 24 department department NOUN ejde-67 426 25 of of ADP ejde-67 426 26 mathematics mathematics PROPN ejde-67 426 27 , , PUNCT ejde-67 426 28 morgan morgan PROPN ejde-67 426 29 state state PROPN ejde-67 426 30 university university PROPN ejde-67 426 31 baltimore baltimore PROPN ejde-67 426 32 , , PUNCT ejde-67 426 33 md md PROPN ejde-67 426 34 21251 21251 NUM ejde-67 426 35 , , PUNCT ejde-67 426 36 usa usa PROPN ejde-67 426 37 email email NOUN ejde-67 426 38 address address NOUN ejde-67 426 39 : : PUNCT ejde-67 426 40 gaston.nguerekata@morgan.edu gaston.nguerekata@morgan.edu NOUN ejde-67 426 41 1 1 NUM ejde-67 426 42 . . PUNCT ejde-67 427 1 introduction introduction NOUN ejde-67 427 2 2 2 NUM ejde-67 427 3 . . PUNCT ejde-67 427 4 variation variation NOUN ejde-67 427 5 of of ADP ejde-67 427 6 constants constant NOUN ejde-67 427 7 formula formula NOUN ejde-67 427 8 and and CCONJ ejde-67 427 9 spectral spectral ADJ ejde-67 427 10 decomposition decomposition NOUN ejde-67 427 11 3 3 NUM ejde-67 427 12 . . PUNCT ejde-67 427 13 reduction reduction NOUN ejde-67 427 14 principle principle NOUN ejde-67 427 15 4 4 NUM ejde-67 427 16 . . PUNCT ejde-67 427 17 almost almost ADV ejde-67 427 18 periodicity periodicity NOUN ejde-67 427 19 and and CCONJ ejde-67 427 20 almost almost ADV ejde-67 427 21 automorphy automorphy NOUN ejde-67 427 22 5 5 NUM ejde-67 427 23 . . PUNCT ejde-67 427 24 existence existence NOUN ejde-67 427 25 of of ADP ejde-67 427 26 almost almost ADV ejde-67 427 27 automorphic automorphic ADJ ejde-67 427 28 and and CCONJ ejde-67 427 29 almost almost ADV ejde-67 427 30 periodic periodic ADJ ejde-67 427 31 solutions solution NOUN ejde-67 427 32 6 6 NUM ejde-67 427 33 . . PUNCT ejde-67 428 1 application application NOUN ejde-67 428 2 7 7 NUM ejde-67 428 3 . . PUNCT ejde-67 428 4 conclusions conclusion NOUN ejde-67 428 5 and and CCONJ ejde-67 428 6 discussion discussion NOUN ejde-67 428 7 acknowledgments acknowledgment NOUN ejde-67 428 8 references reference NOUN