id sid tid token lemma pos american_scientific_journal-4764 1 1 131 131 NUM american_scientific_journal-4764 1 2 american american PROPN american_scientific_journal-4764 1 3 scientific scientific ADJ american_scientific_journal-4764 1 4 research research NOUN american_scientific_journal-4764 1 5 journal journal NOUN american_scientific_journal-4764 1 6 for for ADP american_scientific_journal-4764 1 7 engineering engineering NOUN american_scientific_journal-4764 1 8 , , PUNCT american_scientific_journal-4764 1 9 technology technology NOUN american_scientific_journal-4764 1 10 , , PUNCT american_scientific_journal-4764 1 11 and and CCONJ american_scientific_journal-4764 1 12 sciences science NOUN american_scientific_journal-4764 1 13 ( ( PUNCT american_scientific_journal-4764 1 14 asrjets asrjet NOUN american_scientific_journal-4764 1 15 ) ) PUNCT american_scientific_journal-4764 1 16 issn issn PROPN american_scientific_journal-4764 1 17 ( ( PUNCT american_scientific_journal-4764 1 18 print print NOUN american_scientific_journal-4764 1 19 ) ) PUNCT american_scientific_journal-4764 1 20 2313 2313 NUM american_scientific_journal-4764 1 21 - - SYM american_scientific_journal-4764 1 22 4410 4410 NUM american_scientific_journal-4764 1 23 , , PUNCT american_scientific_journal-4764 1 24 issn issn PROPN american_scientific_journal-4764 1 25 ( ( PUNCT american_scientific_journal-4764 1 26 online online ADJ american_scientific_journal-4764 1 27 ) ) PUNCT american_scientific_journal-4764 1 28 2313 2313 NUM american_scientific_journal-4764 1 29 - - SYM american_scientific_journal-4764 1 30 4402 4402 NUM american_scientific_journal-4764 1 31 Β© Β© PROPN american_scientific_journal-4764 1 32 global global ADJ american_scientific_journal-4764 1 33 society society NOUN american_scientific_journal-4764 1 34 of of ADP american_scientific_journal-4764 1 35 scientific scientific ADJ american_scientific_journal-4764 1 36 research research NOUN american_scientific_journal-4764 1 37 and and CCONJ american_scientific_journal-4764 1 38 researchers researcher NOUN american_scientific_journal-4764 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ NOUN american_scientific_journal-4764 1 40 approximation approximation NOUN american_scientific_journal-4764 1 41 theory theory NOUN american_scientific_journal-4764 1 42 on on ADP american_scientific_journal-4764 1 43 summability summability NOUN american_scientific_journal-4764 1 44 of of ADP american_scientific_journal-4764 1 45 fourier fourier PROPN american_scientific_journal-4764 1 46 series series PROPN american_scientific_journal-4764 1 47 sanjay sanjay PROPN american_scientific_journal-4764 1 48 mukherjeea mukherjeea PROPN american_scientific_journal-4764 1 49 * * PROPN american_scientific_journal-4764 1 50 , , PUNCT american_scientific_journal-4764 1 51 a a DET american_scientific_journal-4764 1 52 j j PROPN american_scientific_journal-4764 1 53 khanb khanb PROPN american_scientific_journal-4764 1 54 aresearch aresearch PROPN american_scientific_journal-4764 1 55 scholor scholor PROPN american_scientific_journal-4764 1 56 , , PUNCT american_scientific_journal-4764 1 57 mats mat VERB american_scientific_journal-4764 1 58 university university NOUN american_scientific_journal-4764 1 59 , , PUNCT american_scientific_journal-4764 1 60 raipur(c.g raipur(c.g PROPN american_scientific_journal-4764 1 61 . . PUNCT american_scientific_journal-4764 1 62 ) ) PUNCT american_scientific_journal-4764 2 1 , , PUNCT american_scientific_journal-4764 2 2 india india PROPN american_scientific_journal-4764 2 3 bmats bmat VERB american_scientific_journal-4764 2 4 university university NOUN american_scientific_journal-4764 2 5 , , PUNCT american_scientific_journal-4764 2 6 raipur(c.g raipur(c.g PROPN american_scientific_journal-4764 2 7 . . PUNCT american_scientific_journal-4764 2 8 ) ) PUNCT american_scientific_journal-4764 2 9 , , PUNCT american_scientific_journal-4764 2 10 india india PROPN american_scientific_journal-4764 2 11 aemail aemail NOUN american_scientific_journal-4764 2 12 : : PUNCT american_scientific_journal-4764 2 13 sanjaymukherjeeruma@gmail.com sanjaymukherjeeruma@gmail.com X american_scientific_journal-4764 2 14 bemail bemail NOUN american_scientific_journal-4764 2 15 : : PUNCT american_scientific_journal-4764 2 16 khanaj@matsuniversity.ac.in khanaj@matsuniversity.ac.in PROPN american_scientific_journal-4764 2 17 abstract abstract VERB american_scientific_journal-4764 2 18 the the DET american_scientific_journal-4764 2 19 results result NOUN american_scientific_journal-4764 2 20 of of ADP american_scientific_journal-4764 2 21 chandra chandra PROPN american_scientific_journal-4764 2 22 to to PART american_scientific_journal-4764 2 23 ( ( PUNCT american_scientific_journal-4764 2 24 e e NOUN american_scientific_journal-4764 2 25 , , PUNCT american_scientific_journal-4764 2 26 c c NOUN american_scientific_journal-4764 2 27 ) ) PUNCT american_scientific_journal-4764 2 28 means mean VERB american_scientific_journal-4764 2 29 u.k.shrivastava u.k.shrivastava NOUN american_scientific_journal-4764 2 30 and and CCONJ american_scientific_journal-4764 2 31 s.k.verma s.k.verma NOUN american_scientific_journal-4764 2 32 have have AUX american_scientific_journal-4764 2 33 proved prove VERB american_scientific_journal-4764 2 34 the the DET american_scientific_journal-4764 2 35 following follow VERB american_scientific_journal-4764 2 36 theorem theorem ADJ american_scientific_journal-4764 2 37 theorem theorem NOUN american_scientific_journal-4764 2 38 : : PUNCT american_scientific_journal-4764 2 39 let let VERB american_scientific_journal-4764 2 40 𝑓𝑓 𝑓𝑓 ADP american_scientific_journal-4764 2 41 ∈ ∈ PROPN american_scientific_journal-4764 2 42 𝐢𝐢2πœ‹πœ‹ 𝐢𝐢2πœ‹πœ‹ NOUN american_scientific_journal-4764 2 43 ∩ ∩ NOUN american_scientific_journal-4764 2 44 𝐿𝐿𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿𝐿𝐿 ADJ american_scientific_journal-4764 2 45 ∝ ∝ PROPN american_scientific_journal-4764 2 46 , , PUNCT american_scientific_journal-4764 2 47 0 0 NUM american_scientific_journal-4764 2 48 < < X american_scientific_journal-4764 2 49 βˆβ‰€ βˆβ‰€ X american_scientific_journal-4764 3 1 1 1 NUM american_scientific_journal-4764 3 2 . . PUNCT american_scientific_journal-4764 4 1 then then ADV american_scientific_journal-4764 4 2 ‖𝑑𝑑𝑛𝑛𝑐𝑐 ‖𝑑𝑑𝑛𝑛𝑐𝑐 X american_scientific_journal-4764 4 3 βˆ’ βˆ’ NOUN american_scientific_journal-4764 4 4 𝑓𝑓‖ 𝑓𝑓‖ PROPN american_scientific_journal-4764 4 5 = = SYM american_scientific_journal-4764 4 6 π‘œπ‘œ π‘œπ‘œ PROPN american_scientific_journal-4764 4 7 οΏ½ οΏ½ PROPN american_scientific_journal-4764 4 8 π‘›π‘›βˆ’βˆ π‘›π‘›βˆ’βˆ PROPN american_scientific_journal-4764 4 9 2 2 NUM american_scientific_journal-4764 4 10 οΏ½ οΏ½ PROPN american_scientific_journal-4764 4 11 οΏ½ οΏ½ PROPN american_scientific_journal-4764 4 12 , , PUNCT american_scientific_journal-4764 4 13 where where SCONJ american_scientific_journal-4764 4 14 𝑑𝑑𝑛𝑛𝑐𝑐(𝑓𝑓 𝑑𝑑𝑛𝑛𝑐𝑐(𝑓𝑓 NOUN american_scientific_journal-4764 4 15 ; ; PUNCT american_scientific_journal-4764 4 16 π‘₯π‘₯ π‘₯π‘₯ X american_scientific_journal-4764 4 17 ) ) PUNCT american_scientific_journal-4764 4 18 is be AUX american_scientific_journal-4764 4 19 nth nth NOUN american_scientific_journal-4764 4 20 ( ( PUNCT american_scientific_journal-4764 4 21 e e NOUN american_scientific_journal-4764 4 22 , , PUNCT american_scientific_journal-4764 4 23 c c NOUN american_scientific_journal-4764 4 24 ) ) PUNCT american_scientific_journal-4764 4 25 means mean NOUN american_scientific_journal-4764 4 26 of of ADP american_scientific_journal-4764 4 27 fourier fourier ADJ american_scientific_journal-4764 4 28 series series NOUN american_scientific_journal-4764 4 29 of of ADP american_scientific_journal-4764 4 30 f f PROPN american_scientific_journal-4764 4 31 at at ADP american_scientific_journal-4764 4 32 x. x. NOUN american_scientific_journal-4764 4 33 in in ADP american_scientific_journal-4764 4 34 this this DET american_scientific_journal-4764 4 35 paper paper NOUN american_scientific_journal-4764 4 36 we we PRON american_scientific_journal-4764 4 37 obtain obtain VERB american_scientific_journal-4764 4 38 the the DET american_scientific_journal-4764 4 39 fourier fourier NOUN american_scientific_journal-4764 4 40 series series NOUN american_scientific_journal-4764 4 41 by by ADP american_scientific_journal-4764 4 42 ( ( PUNCT american_scientific_journal-4764 4 43 n n CCONJ american_scientific_journal-4764 4 44 , , PUNCT american_scientific_journal-4764 4 45 p p X american_scientific_journal-4764 4 46 , , PUNCT american_scientific_journal-4764 4 47 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 4 48 ) ) PUNCT american_scientific_journal-4764 4 49 which which PRON american_scientific_journal-4764 4 50 is be AUX american_scientific_journal-4764 4 51 the the DET american_scientific_journal-4764 4 52 analogues analogue NOUN american_scientific_journal-4764 4 53 to to ADP american_scientific_journal-4764 4 54 the the DET american_scientific_journal-4764 4 55 ( ( PUNCT american_scientific_journal-4764 4 56 e e NOUN american_scientific_journal-4764 4 57 , , PUNCT american_scientific_journal-4764 4 58 c c NOUN american_scientific_journal-4764 4 59 ) ) PUNCT american_scientific_journal-4764 4 60 means mean VERB american_scientific_journal-4764 4 61 given give VERB american_scientific_journal-4764 4 62 above above ADV american_scientific_journal-4764 4 63 .the .the NOUN american_scientific_journal-4764 4 64 theorem theorem NOUN american_scientific_journal-4764 4 65 is be AUX american_scientific_journal-4764 4 66 as as SCONJ american_scientific_journal-4764 4 67 follows follow NOUN american_scientific_journal-4764 4 68 theorem theorem ADJ american_scientific_journal-4764 4 69 : : PUNCT american_scientific_journal-4764 4 70 let let VERB american_scientific_journal-4764 4 71 { { PUNCT american_scientific_journal-4764 4 72 𝐿𝐿𝑛𝑛 𝐿𝐿𝑛𝑛 VERB american_scientific_journal-4764 4 73 } } PUNCT american_scientific_journal-4764 4 74 and and CCONJ american_scientific_journal-4764 4 75 { { PUNCT american_scientific_journal-4764 4 76 π‘žπ‘žπ‘›π‘› π‘žπ‘žπ‘›π‘› NOUN american_scientific_journal-4764 4 77 } } PUNCT american_scientific_journal-4764 4 78 be be AUX american_scientific_journal-4764 4 79 the the DET american_scientific_journal-4764 4 80 positive positive ADJ american_scientific_journal-4764 4 81 monotonic monotonic NOUN american_scientific_journal-4764 4 82 , , PUNCT american_scientific_journal-4764 4 83 non non X american_scientific_journal-4764 4 84 increasing increase VERB american_scientific_journal-4764 4 85 sequence sequence NOUN american_scientific_journal-4764 4 86 of of ADP american_scientific_journal-4764 4 87 real real ADJ american_scientific_journal-4764 4 88 numbers number NOUN american_scientific_journal-4764 4 89 be be AUX american_scientific_journal-4764 4 90 summable summable ADJ american_scientific_journal-4764 4 91 ( ( PUNCT american_scientific_journal-4764 4 92 n n CCONJ american_scientific_journal-4764 4 93 , , PUNCT american_scientific_journal-4764 4 94 p p X american_scientific_journal-4764 4 95 , , PUNCT american_scientific_journal-4764 4 96 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 4 97 ) ) PUNCT american_scientific_journal-4764 4 98 to to ADP american_scientific_journal-4764 4 99 f(x f(x PROPN american_scientific_journal-4764 4 100 ) ) PUNCT american_scientific_journal-4764 4 101 at at ADP american_scientific_journal-4764 4 102 the the DET american_scientific_journal-4764 4 103 point point NOUN american_scientific_journal-4764 4 104 t t NOUN american_scientific_journal-4764 4 105 = = NOUN american_scientific_journal-4764 4 106 x x X american_scientific_journal-4764 4 107 is be AUX american_scientific_journal-4764 4 108 𝑑𝑑𝑁𝑁 𝑑𝑑𝑁𝑁 NOUN american_scientific_journal-4764 4 109 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 NUM american_scientific_journal-4764 4 110 βˆ’ βˆ’ PROPN american_scientific_journal-4764 4 111 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 4 112 ) ) PUNCT american_scientific_journal-4764 4 113 = = SYM american_scientific_journal-4764 4 114 π‘œπ‘œ(1 π‘œπ‘œ(1 NOUN american_scientific_journal-4764 4 115 ) ) PUNCT american_scientific_journal-4764 4 116 keywords keyword NOUN american_scientific_journal-4764 4 117 : : PUNCT american_scientific_journal-4764 4 118 fourier fourier ADJ american_scientific_journal-4764 4 119 series series NOUN american_scientific_journal-4764 4 120 ; ; PUNCT american_scientific_journal-4764 4 121 borel borel PROPN american_scientific_journal-4764 4 122 means mean VERB american_scientific_journal-4764 4 123 ; ; PUNCT american_scientific_journal-4764 4 124 lebesgue lebesgue NOUN american_scientific_journal-4764 4 125 series series NOUN american_scientific_journal-4764 4 126 . . PUNCT american_scientific_journal-4764 5 1 ---------------------------------------------------------------------- ---------------------------------------------------------------------- PUNCT american_scientific_journal-4764 6 1 * * PUNCT american_scientific_journal-4764 6 2 corresponding correspond VERB american_scientific_journal-4764 6 3 author author NOUN american_scientific_journal-4764 6 4 . . PUNCT american_scientific_journal-4764 7 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-4764 7 2 american american PROPN american_scientific_journal-4764 7 3 scientific scientific PROPN american_scientific_journal-4764 7 4 research research NOUN american_scientific_journal-4764 7 5 journal journal NOUN american_scientific_journal-4764 7 6 for for ADP american_scientific_journal-4764 7 7 engineering engineering NOUN american_scientific_journal-4764 7 8 , , PUNCT american_scientific_journal-4764 7 9 technology technology NOUN american_scientific_journal-4764 7 10 , , PUNCT american_scientific_journal-4764 7 11 and and CCONJ american_scientific_journal-4764 7 12 sciences science NOUN american_scientific_journal-4764 7 13 ( ( PUNCT american_scientific_journal-4764 7 14 asrjets asrjet NOUN american_scientific_journal-4764 7 15 ) ) PUNCT american_scientific_journal-4764 7 16 ( ( PUNCT american_scientific_journal-4764 7 17 2019 2019 NUM american_scientific_journal-4764 7 18 ) ) PUNCT american_scientific_journal-4764 7 19 volume volume NOUN american_scientific_journal-4764 7 20 54 54 NUM american_scientific_journal-4764 7 21 , , PUNCT american_scientific_journal-4764 7 22 no no DET american_scientific_journal-4764 7 23 1 1 NUM american_scientific_journal-4764 7 24 , , PUNCT american_scientific_journal-4764 7 25 pp pp ADV american_scientific_journal-4764 7 26 131 131 NUM american_scientific_journal-4764 7 27 - - SYM american_scientific_journal-4764 7 28 136 136 NUM american_scientific_journal-4764 7 29 132 132 NUM american_scientific_journal-4764 7 30 1 1 NUM american_scientific_journal-4764 7 31 . . PUNCT american_scientific_journal-4764 8 1 introduction introduction NOUN american_scientific_journal-4764 8 2 let let VERB american_scientific_journal-4764 8 3 { { PUNCT american_scientific_journal-4764 8 4 𝐿𝐿𝑛𝑛 𝐿𝐿𝑛𝑛 VERB american_scientific_journal-4764 8 5 } } PUNCT american_scientific_journal-4764 8 6 and and CCONJ american_scientific_journal-4764 8 7 { { PUNCT american_scientific_journal-4764 8 8 π‘žπ‘žπ‘›π‘› π‘žπ‘žπ‘›π‘› NOUN american_scientific_journal-4764 8 9 } } PUNCT american_scientific_journal-4764 8 10 be be AUX american_scientific_journal-4764 8 11 the the DET american_scientific_journal-4764 8 12 sequences sequence NOUN american_scientific_journal-4764 8 13 of of ADP american_scientific_journal-4764 8 14 constants constant NOUN american_scientific_journal-4764 8 15 , , PUNCT american_scientific_journal-4764 8 16 real real ADJ american_scientific_journal-4764 8 17 or or CCONJ american_scientific_journal-4764 8 18 complex complex ADJ american_scientific_journal-4764 8 19 , , PUNCT american_scientific_journal-4764 9 1 such such ADJ american_scientific_journal-4764 9 2 that that SCONJ american_scientific_journal-4764 9 3 𝑃𝑃𝑛𝑛 𝑃𝑃𝑛𝑛 PROPN american_scientific_journal-4764 9 4 = = PUNCT american_scientific_journal-4764 9 5 𝐿𝐿1 𝐿𝐿1 PROPN american_scientific_journal-4764 9 6 + + PROPN american_scientific_journal-4764 9 7 𝐿𝐿2 𝐿𝐿2 PROPN american_scientific_journal-4764 9 8 + + CCONJ american_scientific_journal-4764 9 9 𝐿𝐿3 𝐿𝐿3 PROPN american_scientific_journal-4764 9 10 + + CCONJ american_scientific_journal-4764 9 11 ⋯𝐿𝐿𝑛𝑛 ⋯𝐿𝐿𝑛𝑛 NOUN american_scientific_journal-4764 9 12 = = SYM american_scientific_journal-4764 9 13 οΏ½ οΏ½ PROPN american_scientific_journal-4764 9 14 πΏπΏπ‘Ÿπ‘Ÿ πΏπΏπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 9 15 β†’ β†’ SYM american_scientific_journal-4764 9 16 ∞ ∞ PROPN american_scientific_journal-4764 9 17 , , PUNCT american_scientific_journal-4764 9 18 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 9 19 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 9 20 β†’ β†’ PUNCT american_scientific_journal-4764 9 21 ∞ ∞ PROPN american_scientific_journal-4764 9 22 , , PUNCT american_scientific_journal-4764 9 23 𝑛𝑛 𝑛𝑛 VERB american_scientific_journal-4764 9 24 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 NOUN american_scientific_journal-4764 9 25 𝑄𝑄𝑛𝑛 𝑄𝑄𝑛𝑛 PROPN american_scientific_journal-4764 9 26 = = PRON american_scientific_journal-4764 9 27 π‘žπ‘ž1 π‘žπ‘ž1 X american_scientific_journal-4764 9 28 + + CCONJ american_scientific_journal-4764 9 29 π‘žπ‘ž2 π‘žπ‘ž2 PROPN american_scientific_journal-4764 9 30 + + CCONJ american_scientific_journal-4764 9 31 π‘žπ‘ž3 π‘žπ‘ž3 PROPN american_scientific_journal-4764 9 32 + + PROPN american_scientific_journal-4764 9 33 β‹―π‘žπ‘žπ‘›π‘› β‹―π‘žπ‘žπ‘›π‘› PROPN american_scientific_journal-4764 9 34 = = SYM american_scientific_journal-4764 9 35 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 9 36 π‘žπ‘žπ‘Ÿπ‘Ÿ π‘žπ‘žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 9 37 β†’ β†’ SYM american_scientific_journal-4764 9 38 ∞ ∞ PROPN american_scientific_journal-4764 9 39 , , PUNCT american_scientific_journal-4764 9 40 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 9 41 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 9 42 β†’ β†’ PUNCT american_scientific_journal-4764 9 43 βˆžπ‘›π‘› βˆžπ‘›π‘› X american_scientific_journal-4764 9 44 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 NUM american_scientific_journal-4764 9 45 , , PUNCT american_scientific_journal-4764 9 46 ( ( PUNCT american_scientific_journal-4764 9 47 1.1 1.1 NUM american_scientific_journal-4764 9 48 ) ) PUNCT american_scientific_journal-4764 9 49 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 NOUN american_scientific_journal-4764 9 50 = = SYM american_scientific_journal-4764 9 51 𝐿𝐿0π‘žπ‘žπ‘›π‘› 𝐿𝐿0π‘žπ‘žπ‘›π‘› PROPN american_scientific_journal-4764 10 1 + + CCONJ american_scientific_journal-4764 10 2 𝐿𝐿1π‘žπ‘žπ‘›π‘›βˆ’1 𝐿𝐿1π‘žπ‘žπ‘›π‘›βˆ’1 NOUN american_scientific_journal-4764 10 3 + + CCONJ american_scientific_journal-4764 10 4 𝐿𝐿3π‘žπ‘žπ‘›π‘›βˆ’2 𝐿𝐿3π‘žπ‘žπ‘›π‘›βˆ’2 PROPN american_scientific_journal-4764 10 5 + + CCONJ american_scientific_journal-4764 10 6 β‹―πΏπΏπ‘›π‘›π‘žπ‘ž0 β‹―πΏπΏπ‘›π‘›π‘žπ‘ž0 X american_scientific_journal-4764 10 7 = = PUNCT american_scientific_journal-4764 10 8 οΏ½ οΏ½ PROPN american_scientific_journal-4764 10 9 πΏπΏπ‘Ÿπ‘Ÿπ‘žπ‘žπ‘›π‘›βˆ’π‘Ÿπ‘Ÿ πΏπΏπ‘Ÿπ‘Ÿπ‘žπ‘žπ‘›π‘›βˆ’π‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 10 10 β†’ β†’ SYM american_scientific_journal-4764 10 11 ∞ ∞ PROPN american_scientific_journal-4764 10 12 , , PUNCT american_scientific_journal-4764 10 13 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 10 14 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 10 15 β†’ β†’ SYM american_scientific_journal-4764 10 16 ∞ ∞ PROPN american_scientific_journal-4764 10 17 𝑛𝑛 𝑛𝑛 X american_scientific_journal-4764 10 18 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PRON american_scientific_journal-4764 10 19 given give VERB american_scientific_journal-4764 10 20 two two NUM american_scientific_journal-4764 10 21 sequences sequence NOUN american_scientific_journal-4764 10 22 { { PUNCT american_scientific_journal-4764 10 23 𝐿𝐿𝑛𝑛 𝐿𝐿𝑛𝑛 NOUN american_scientific_journal-4764 10 24 } } PUNCT american_scientific_journal-4764 10 25 and and CCONJ american_scientific_journal-4764 10 26 { { PUNCT american_scientific_journal-4764 10 27 π‘žπ‘žπ‘›π‘› π‘žπ‘žπ‘›π‘› NOUN american_scientific_journal-4764 10 28 } } PUNCT american_scientific_journal-4764 10 29 convolution convolution NOUN american_scientific_journal-4764 10 30 ( ( PUNCT american_scientific_journal-4764 10 31 𝐿𝐿 𝐿𝐿 PROPN american_scientific_journal-4764 10 32 βˆ— βˆ— NOUN american_scientific_journal-4764 10 33 π‘žπ‘ž π‘žπ‘ž NOUN american_scientific_journal-4764 10 34 ) ) PUNCT american_scientific_journal-4764 10 35 is be AUX american_scientific_journal-4764 10 36 defined define VERB american_scientific_journal-4764 10 37 as as ADP american_scientific_journal-4764 10 38 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 PROPN american_scientific_journal-4764 10 39 = = SYM american_scientific_journal-4764 10 40 ( ( PUNCT american_scientific_journal-4764 10 41 𝐿𝐿 𝐿𝐿 PROPN american_scientific_journal-4764 10 42 β‰  β‰  PROPN american_scientific_journal-4764 10 43 π‘žπ‘ž)𝑛𝑛 π‘žπ‘ž)𝑛𝑛 PROPN american_scientific_journal-4764 10 44 = = SYM american_scientific_journal-4764 10 45 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 10 46 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘›π‘› πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘›π‘› PROPN american_scientific_journal-4764 10 47 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 10 48 π‘žπ‘žπ‘Ÿπ‘Ÿ π‘žπ‘žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 10 49 ( ( PUNCT american_scientific_journal-4764 10 50 1.2 1.2 NUM american_scientific_journal-4764 10 51 ) ) PUNCT american_scientific_journal-4764 10 52 let let VERB american_scientific_journal-4764 10 53 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 10 54 π‘’π‘’π‘›π‘›βˆž π‘’π‘’π‘›π‘›βˆž NOUN american_scientific_journal-4764 10 55 𝑛𝑛=0 𝑛𝑛=0 NOUN american_scientific_journal-4764 10 56 be be VERB american_scientific_journal-4764 10 57 an an DET american_scientific_journal-4764 10 58 infinite infinite ADJ american_scientific_journal-4764 10 59 series series NOUN american_scientific_journal-4764 10 60 with with ADP american_scientific_journal-4764 10 61 the the DET american_scientific_journal-4764 10 62 sequence sequence NOUN american_scientific_journal-4764 10 63 of of ADP american_scientific_journal-4764 10 64 its its PRON american_scientific_journal-4764 10 65 nth nth NOUN american_scientific_journal-4764 10 66 partial partial ADJ american_scientific_journal-4764 10 67 sums{π‘Žπ‘Žπ‘›π‘› sums{π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-4764 10 68 } } PUNCT american_scientific_journal-4764 10 69 . . PUNCT american_scientific_journal-4764 11 1 we we PRON american_scientific_journal-4764 11 2 write write VERB american_scientific_journal-4764 11 3 𝑑𝑑𝑛𝑛 𝑑𝑑𝑛𝑛 PROPN american_scientific_journal-4764 11 4 𝑝𝑝,π‘žπ‘ž 𝑝𝑝,π‘žπ‘ž NOUN american_scientific_journal-4764 11 5 = = SYM american_scientific_journal-4764 11 6 1 1 NUM american_scientific_journal-4764 11 7 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 NOUN american_scientific_journal-4764 11 8 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 11 9 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿπ‘›π‘› πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿπ‘›π‘› PROPN american_scientific_journal-4764 11 10 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 11 11 ( ( PUNCT american_scientific_journal-4764 11 12 1.3 1.3 NUM american_scientific_journal-4764 11 13 ) ) PUNCT american_scientific_journal-4764 11 14 if if SCONJ american_scientific_journal-4764 11 15 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 ADP american_scientific_journal-4764 11 16 β‰  β‰  PROPN american_scientific_journal-4764 11 17 0 0 NUM american_scientific_journal-4764 11 18 , , PUNCT american_scientific_journal-4764 11 19 for for ADP american_scientific_journal-4764 11 20 all all DET american_scientific_journal-4764 11 21 n n CCONJ american_scientific_journal-4764 11 22 , , PUNCT american_scientific_journal-4764 11 23 the the DET american_scientific_journal-4764 11 24 generalized generalized ADJ american_scientific_journal-4764 11 25 norlund norlund ADJ american_scientific_journal-4764 11 26 transform transform NOUN american_scientific_journal-4764 11 27 of of ADP american_scientific_journal-4764 11 28 the the DET american_scientific_journal-4764 11 29 sequence sequence NOUN american_scientific_journal-4764 11 30 { { PUNCT american_scientific_journal-4764 11 31 π‘Žπ‘Žπ‘›π‘› π‘Žπ‘Žπ‘›π‘› PROPN american_scientific_journal-4764 11 32 } } PUNCT american_scientific_journal-4764 11 33 is be AUX american_scientific_journal-4764 11 34 the the DET american_scientific_journal-4764 11 35 sequence sequence NOUN american_scientific_journal-4764 11 36 οΏ½ οΏ½ PROPN american_scientific_journal-4764 11 37 𝑑𝑑𝑛𝑛 𝑑𝑑𝑛𝑛 PROPN american_scientific_journal-4764 11 38 𝑝𝑝.π‘žπ‘ž 𝑝𝑝.π‘žπ‘ž PROPN american_scientific_journal-4764 11 39 οΏ½ οΏ½ PROPN american_scientific_journal-4764 11 40 . . PUNCT american_scientific_journal-4764 12 1 if if SCONJ american_scientific_journal-4764 12 2 𝑑𝑑𝑛𝑛 𝑑𝑑𝑛𝑛 VERB american_scientific_journal-4764 12 3 𝑝𝑝,π‘žπ‘ž 𝑝𝑝,π‘žπ‘ž NOUN american_scientific_journal-4764 12 4 β†’ β†’ SYM american_scientific_journal-4764 12 5 𝑆𝑆 𝑆𝑆 PROPN american_scientific_journal-4764 12 6 , , PUNCT american_scientific_journal-4764 12 7 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 12 8 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 12 9 β†’ β†’ PUNCT american_scientific_journal-4764 12 10 ∞ ∞ PROPN american_scientific_journal-4764 12 11 , , PUNCT american_scientific_journal-4764 12 12 then then ADV american_scientific_journal-4764 12 13 the the DET american_scientific_journal-4764 12 14 series series NOUN american_scientific_journal-4764 12 15 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 12 16 π‘’π‘’π‘›π‘›βˆž π‘’π‘’π‘›π‘›βˆž NOUN american_scientific_journal-4764 12 17 𝑛𝑛=0 𝑛𝑛=0 NOUN american_scientific_journal-4764 12 18 or or CCONJ american_scientific_journal-4764 12 19 sequence sequence NOUN american_scientific_journal-4764 12 20 { { PUNCT american_scientific_journal-4764 12 21 π‘Žπ‘Žπ‘›π‘› π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-4764 12 22 } } PUNCT american_scientific_journal-4764 12 23 is be AUX american_scientific_journal-4764 12 24 summable summable ADJ american_scientific_journal-4764 12 25 to to ADP american_scientific_journal-4764 12 26 s s PRON american_scientific_journal-4764 12 27 by by ADP american_scientific_journal-4764 12 28 𝑆𝑆𝑛𝑛 𝑆𝑆𝑛𝑛 PROPN american_scientific_journal-4764 12 29 β†’ β†’ SYM american_scientific_journal-4764 12 30 𝑆𝑆(𝑁𝑁 𝑆𝑆(𝑁𝑁 PROPN american_scientific_journal-4764 12 31 , , PUNCT american_scientific_journal-4764 12 32 𝐿𝐿 𝐿𝐿 PROPN american_scientific_journal-4764 12 33 , , PUNCT american_scientific_journal-4764 12 34 π‘žπ‘ž π‘žπ‘ž PROPN american_scientific_journal-4764 12 35 ) ) PUNCT american_scientific_journal-4764 12 36 ( ( PUNCT american_scientific_journal-4764 12 37 1.4 1.4 NUM american_scientific_journal-4764 12 38 ) ) PUNCT american_scientific_journal-4764 12 39 the the DET american_scientific_journal-4764 12 40 necessary necessary ADJ american_scientific_journal-4764 12 41 and and CCONJ american_scientific_journal-4764 12 42 sufficient sufficient ADJ american_scientific_journal-4764 12 43 conditions condition NOUN american_scientific_journal-4764 12 44 for for ADP american_scientific_journal-4764 12 45 ( ( PUNCT american_scientific_journal-4764 12 46 n n CCONJ american_scientific_journal-4764 12 47 , , PUNCT american_scientific_journal-4764 12 48 p p X american_scientific_journal-4764 12 49 , , PUNCT american_scientific_journal-4764 12 50 q q NOUN american_scientific_journal-4764 12 51 ) ) PUNCT american_scientific_journal-4764 12 52 method method NOUN american_scientific_journal-4764 12 53 to to PART american_scientific_journal-4764 12 54 be be AUX american_scientific_journal-4764 12 55 regular regular ADJ american_scientific_journal-4764 12 56 are be AUX american_scientific_journal-4764 12 57 βˆ‘ βˆ‘ ADV american_scientific_journal-4764 12 58 |πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿ| |πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿ| PUNCT american_scientific_journal-4764 12 59 = = NOUN american_scientific_journal-4764 12 60 π‘œπ‘œ(|𝑅𝑅𝑛𝑛|)𝑛𝑛 π‘œπ‘œ(|𝑅𝑅𝑛𝑛|)𝑛𝑛 X american_scientific_journal-4764 12 61 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PRON american_scientific_journal-4764 12 62 ( ( PUNCT american_scientific_journal-4764 12 63 1.5 1.5 NUM american_scientific_journal-4764 12 64 ) ) PUNCT american_scientific_journal-4764 12 65 and and CCONJ american_scientific_journal-4764 12 66 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿ πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 12 67 = = SYM american_scientific_journal-4764 12 68 π‘œπ‘œ(|𝑅𝑅𝑛𝑛| π‘œπ‘œ(|𝑅𝑅𝑛𝑛| PROPN american_scientific_journal-4764 12 69 ) ) PUNCT american_scientific_journal-4764 12 70 , , PUNCT american_scientific_journal-4764 12 71 as as ADP american_scientific_journal-4764 12 72 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 12 73 β†’ β†’ SYM american_scientific_journal-4764 12 74 ∞ ∞ PROPN american_scientific_journal-4764 12 75 for for ADP american_scientific_journal-4764 12 76 every every DET american_scientific_journal-4764 12 77 fixed fix VERB american_scientific_journal-4764 12 78 π‘˜π‘˜ π‘˜π‘˜ NOUN american_scientific_journal-4764 12 79 β‰₯ β‰₯ PROPN american_scientific_journal-4764 12 80 0 0 NUM american_scientific_journal-4764 12 81 , , PUNCT american_scientific_journal-4764 12 82 for for ADP american_scientific_journal-4764 12 83 which which PRON american_scientific_journal-4764 12 84 π‘žπ‘žπ‘Ÿπ‘Ÿ π‘žπ‘žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 12 85 β‰  β‰  PROPN american_scientific_journal-4764 12 86 0 0 NUM american_scientific_journal-4764 12 87 𝐸𝐸𝑛𝑛1 𝐸𝐸𝑛𝑛1 NOUN american_scientific_journal-4764 12 88 = = SYM american_scientific_journal-4764 12 89 1 1 NUM american_scientific_journal-4764 12 90 2𝑛𝑛 2𝑛𝑛 NOUN american_scientific_journal-4764 12 91 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 12 92 οΏ½ οΏ½ PROPN american_scientific_journal-4764 12 93 π‘›π‘›π‘Ÿπ‘Ÿ π‘›π‘›π‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 12 94 οΏ½ οΏ½ PROPN american_scientific_journal-4764 12 95 𝑛𝑛 𝑛𝑛 ADP american_scientific_journal-4764 12 96 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 12 97 π‘Žπ‘Žπ‘Ÿπ‘Ÿ π‘Žπ‘Žπ‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 12 98 ( ( PUNCT american_scientific_journal-4764 12 99 1.6 1.6 NUM american_scientific_journal-4764 12 100 ) ) PUNCT american_scientific_journal-4764 12 101 if if SCONJ american_scientific_journal-4764 12 102 𝐸𝐸𝑛𝑛1 𝐸𝐸𝑛𝑛1 NOUN american_scientific_journal-4764 12 103 β†’ β†’ SYM american_scientific_journal-4764 12 104 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4764 12 105 , , PUNCT american_scientific_journal-4764 12 106 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 12 107 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 12 108 β†’ β†’ SYM american_scientific_journal-4764 12 109 ∞ ∞ PROPN american_scientific_journal-4764 12 110 , , PUNCT american_scientific_journal-4764 12 111 then then ADV american_scientific_journal-4764 12 112 the the DET american_scientific_journal-4764 12 113 series series NOUN american_scientific_journal-4764 12 114 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4764 12 115 π‘’π‘’π‘›π‘›βˆž π‘’π‘’π‘›π‘›βˆž PROPN american_scientific_journal-4764 12 116 𝑛𝑛=0 𝑛𝑛=0 NOUN american_scientific_journal-4764 12 117 is be AUX american_scientific_journal-4764 12 118 said say VERB american_scientific_journal-4764 12 119 to to PART american_scientific_journal-4764 12 120 be be AUX american_scientific_journal-4764 12 121 ( ( PUNCT american_scientific_journal-4764 12 122 e,1 e,1 NOUN american_scientific_journal-4764 12 123 ) ) PUNCT american_scientific_journal-4764 12 124 summable summable ADJ american_scientific_journal-4764 12 125 to to ADP american_scientific_journal-4764 12 126 s s PRON american_scientific_journal-4764 12 127 ( ( PUNCT american_scientific_journal-4764 12 128 hardy hardy ADJ american_scientific_journal-4764 12 129 [ [ X american_scientific_journal-4764 12 130 1 1 NUM american_scientific_journal-4764 12 131 ] ] PUNCT american_scientific_journal-4764 12 132 ) ) PUNCT american_scientific_journal-4764 12 133 : : PUNCT american_scientific_journal-4764 12 134 𝑑𝑑𝑛𝑛 𝑑𝑑𝑛𝑛 PROPN american_scientific_journal-4764 12 135 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 PROPN american_scientific_journal-4764 12 136 = = SYM american_scientific_journal-4764 12 137 1 1 NUM american_scientific_journal-4764 12 138 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 NOUN american_scientific_journal-4764 12 139 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4764 12 140 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘ŸπΈπΈπ‘Ÿπ‘Ÿ1𝑛𝑛 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘ŸπΈπΈπ‘Ÿπ‘Ÿ1𝑛𝑛 VERB american_scientific_journal-4764 12 141 π‘Ÿπ‘Ÿβˆ’0 π‘Ÿπ‘Ÿβˆ’0 VERB american_scientific_journal-4764 12 142 = = SYM american_scientific_journal-4764 12 143 1 1 NUM american_scientific_journal-4764 12 144 𝑅𝑅𝑛𝑛 𝑅𝑅𝑛𝑛 NOUN american_scientific_journal-4764 12 145 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 12 146 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿπ‘›π‘› πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿπ‘›π‘› PROPN american_scientific_journal-4764 12 147 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 12 148 1 1 NUM american_scientific_journal-4764 12 149 2π‘˜π‘˜ 2π‘˜π‘˜ NOUN american_scientific_journal-4764 12 150 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 12 151 οΏ½ οΏ½ PROPN american_scientific_journal-4764 12 152 π‘˜π‘˜π‘Ÿπ‘Ÿ π‘˜π‘˜π‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 12 153 οΏ½ οΏ½ PROPN american_scientific_journal-4764 12 154 π‘Žπ‘Žπ‘Ÿπ‘Ÿ π‘Žπ‘Žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 12 155 𝑛𝑛 𝑛𝑛 ADP american_scientific_journal-4764 12 156 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 NOUN american_scientific_journal-4764 12 157 ( ( PUNCT american_scientific_journal-4764 12 158 1.7 1.7 NUM american_scientific_journal-4764 12 159 ) ) PUNCT american_scientific_journal-4764 12 160 if if SCONJ american_scientific_journal-4764 12 161 𝑇𝑇𝑛𝑛 𝑇𝑇𝑛𝑛 PROPN american_scientific_journal-4764 12 162 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 PROPN american_scientific_journal-4764 12 163 β†’ β†’ SYM american_scientific_journal-4764 12 164 ∞,π‘Žπ‘Žπ‘Žπ‘Ž ∞,π‘Žπ‘Žπ‘Žπ‘Ž X american_scientific_journal-4764 12 165 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 12 166 β†’ β†’ SYM american_scientific_journal-4764 12 167 ∞ ∞ PROPN american_scientific_journal-4764 12 168 , , PUNCT american_scientific_journal-4764 12 169 then then ADV american_scientific_journal-4764 12 170 we we PRON american_scientific_journal-4764 12 171 say say VERB american_scientific_journal-4764 12 172 that that SCONJ american_scientific_journal-4764 12 173 the the DET american_scientific_journal-4764 12 174 series series NOUN american_scientific_journal-4764 12 175 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4764 12 176 π‘’π‘’π‘›π‘›βˆž π‘’π‘’π‘›π‘›βˆž NOUN american_scientific_journal-4764 12 177 𝑛𝑛=0 𝑛𝑛=0 NOUN american_scientific_journal-4764 12 178 or or CCONJ american_scientific_journal-4764 12 179 the the DET american_scientific_journal-4764 12 180 sequence sequence NOUN american_scientific_journal-4764 12 181 { { PUNCT american_scientific_journal-4764 12 182 π‘Žπ‘Žπ‘›π‘› π‘Žπ‘Žπ‘›π‘› NOUN american_scientific_journal-4764 12 183 } } PUNCT american_scientific_journal-4764 12 184 is be AUX american_scientific_journal-4764 12 185 summable summable ADJ american_scientific_journal-4764 12 186 to to ADP american_scientific_journal-4764 12 187 s s PRON american_scientific_journal-4764 12 188 by by ADP american_scientific_journal-4764 12 189 american american ADJ american_scientific_journal-4764 12 190 scientific scientific PROPN american_scientific_journal-4764 12 191 research research PROPN american_scientific_journal-4764 12 192 journal journal NOUN american_scientific_journal-4764 12 193 for for ADP american_scientific_journal-4764 12 194 engineering engineering NOUN american_scientific_journal-4764 12 195 , , PUNCT american_scientific_journal-4764 12 196 technology technology NOUN american_scientific_journal-4764 12 197 , , PUNCT american_scientific_journal-4764 12 198 and and CCONJ american_scientific_journal-4764 12 199 sciences science NOUN american_scientific_journal-4764 12 200 ( ( PUNCT american_scientific_journal-4764 12 201 asrjets asrjet NOUN american_scientific_journal-4764 12 202 ) ) PUNCT american_scientific_journal-4764 12 203 ( ( PUNCT american_scientific_journal-4764 12 204 2019 2019 NUM american_scientific_journal-4764 12 205 ) ) PUNCT american_scientific_journal-4764 12 206 volume volume NOUN american_scientific_journal-4764 12 207 54 54 NUM american_scientific_journal-4764 12 208 , , PUNCT american_scientific_journal-4764 12 209 no no DET american_scientific_journal-4764 12 210 1 1 NUM american_scientific_journal-4764 12 211 , , PUNCT american_scientific_journal-4764 12 212 pp pp ADV american_scientific_journal-4764 12 213 131 131 NUM american_scientific_journal-4764 12 214 - - SYM american_scientific_journal-4764 12 215 136 136 NUM american_scientific_journal-4764 12 216 133 133 NUM american_scientific_journal-4764 12 217 ( ( PUNCT american_scientific_journal-4764 12 218 n n CCONJ american_scientific_journal-4764 12 219 , , PUNCT american_scientific_journal-4764 12 220 p p X american_scientific_journal-4764 12 221 , , PUNCT american_scientific_journal-4764 12 222 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 12 223 ) ) PUNCT american_scientific_journal-4764 12 224 summability summability NOUN american_scientific_journal-4764 12 225 method method NOUN american_scientific_journal-4764 12 226 . . PUNCT american_scientific_journal-4764 13 1 2 2 X american_scientific_journal-4764 13 2 . . X american_scientific_journal-4764 13 3 structure structure NOUN american_scientific_journal-4764 13 4 2 2 NUM american_scientific_journal-4764 13 5 . . NOUN american_scientific_journal-4764 13 6 degree degree NOUN american_scientific_journal-4764 13 7 of of ADP american_scientific_journal-4764 13 8 approximation approximation NOUN american_scientific_journal-4764 13 9 by by ADP american_scientific_journal-4764 13 10 borel borel PROPN american_scientific_journal-4764 13 11 means means PROPN american_scientific_journal-4764 13 12 and and CCONJ american_scientific_journal-4764 13 13 ( ( PUNCT american_scientific_journal-4764 13 14 e e NOUN american_scientific_journal-4764 13 15 , , PUNCT american_scientific_journal-4764 13 16 q q NOUN american_scientific_journal-4764 13 17 ) ) PUNCT american_scientific_journal-4764 13 18 means mean NOUN american_scientific_journal-4764 13 19 were be AUX american_scientific_journal-4764 13 20 obtained obtain VERB american_scientific_journal-4764 13 21 by by ADP american_scientific_journal-4764 13 22 chandra chandra NOUN american_scientific_journal-4764 13 23 [ [ X american_scientific_journal-4764 13 24 4 4 NUM american_scientific_journal-4764 13 25 ] ] PUNCT american_scientific_journal-4764 13 26 and and CCONJ american_scientific_journal-4764 13 27 [ [ X american_scientific_journal-4764 13 28 5 5 NUM american_scientific_journal-4764 13 29 ] ] PUNCT american_scientific_journal-4764 13 30 respectively respectively ADV american_scientific_journal-4764 13 31 .extending .extende VERB american_scientific_journal-4764 13 32 the the DET american_scientific_journal-4764 13 33 results result NOUN american_scientific_journal-4764 13 34 of of ADP american_scientific_journal-4764 13 35 chandra chandra PROPN american_scientific_journal-4764 13 36 to to PART american_scientific_journal-4764 13 37 ( ( PUNCT american_scientific_journal-4764 13 38 e e NOUN american_scientific_journal-4764 13 39 , , PUNCT american_scientific_journal-4764 13 40 c c NOUN american_scientific_journal-4764 13 41 ) ) PUNCT american_scientific_journal-4764 13 42 means mean VERB american_scientific_journal-4764 13 43 u.k.shrivastava u.k.shrivastava NOUN american_scientific_journal-4764 13 44 and and CCONJ american_scientific_journal-4764 13 45 s.k.verma[9 s.k.verma[9 PROPN american_scientific_journal-4764 13 46 ] ] PUNCT american_scientific_journal-4764 13 47 have have AUX american_scientific_journal-4764 13 48 proved prove VERB american_scientific_journal-4764 13 49 the the DET american_scientific_journal-4764 13 50 following follow VERB american_scientific_journal-4764 13 51 theorem theorem ADJ american_scientific_journal-4764 13 52 theorem theorem NOUN american_scientific_journal-4764 13 53 : : PUNCT american_scientific_journal-4764 13 54 let let VERB american_scientific_journal-4764 13 55 𝑓𝑓 𝑓𝑓 ADP american_scientific_journal-4764 13 56 ∈ ∈ PROPN american_scientific_journal-4764 13 57 𝐢𝐢2πœ‹πœ‹ 𝐢𝐢2πœ‹πœ‹ NOUN american_scientific_journal-4764 13 58 ∩ ∩ NOUN american_scientific_journal-4764 13 59 𝐿𝐿𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿𝐿𝐿 ADJ american_scientific_journal-4764 13 60 ∝ ∝ PROPN american_scientific_journal-4764 13 61 , , PUNCT american_scientific_journal-4764 13 62 0 0 NUM american_scientific_journal-4764 13 63 < < X american_scientific_journal-4764 13 64 βˆβ‰€ βˆβ‰€ X american_scientific_journal-4764 14 1 1 1 NUM american_scientific_journal-4764 14 2 . . PUNCT american_scientific_journal-4764 15 1 then then ADV american_scientific_journal-4764 15 2 ‖𝑑𝑑𝑛𝑛𝑐𝑐 ‖𝑑𝑑𝑛𝑛𝑐𝑐 X american_scientific_journal-4764 15 3 βˆ’ βˆ’ NOUN american_scientific_journal-4764 15 4 𝑓𝑓‖ 𝑓𝑓‖ PROPN american_scientific_journal-4764 15 5 = = SYM american_scientific_journal-4764 15 6 π‘œπ‘œ π‘œπ‘œ PROPN american_scientific_journal-4764 15 7 οΏ½ οΏ½ PROPN american_scientific_journal-4764 15 8 π‘›π‘›βˆ’βˆ π‘›π‘›βˆ’βˆ PROPN american_scientific_journal-4764 15 9 2 2 NUM american_scientific_journal-4764 15 10 οΏ½ οΏ½ PROPN american_scientific_journal-4764 15 11 οΏ½ οΏ½ PROPN american_scientific_journal-4764 15 12 , , PUNCT american_scientific_journal-4764 15 13 where where SCONJ american_scientific_journal-4764 15 14 𝑑𝑑𝑛𝑛𝑐𝑐(𝑓𝑓 𝑑𝑑𝑛𝑛𝑐𝑐(𝑓𝑓 NOUN american_scientific_journal-4764 15 15 ; ; PUNCT american_scientific_journal-4764 15 16 π‘₯π‘₯ π‘₯π‘₯ X american_scientific_journal-4764 15 17 ) ) PUNCT american_scientific_journal-4764 15 18 is be AUX american_scientific_journal-4764 15 19 nth nth NOUN american_scientific_journal-4764 15 20 ( ( PUNCT american_scientific_journal-4764 15 21 e e NOUN american_scientific_journal-4764 15 22 , , PUNCT american_scientific_journal-4764 15 23 c c NOUN american_scientific_journal-4764 15 24 ) ) PUNCT american_scientific_journal-4764 15 25 means mean NOUN american_scientific_journal-4764 15 26 of of ADP american_scientific_journal-4764 15 27 fourier fourier ADJ american_scientific_journal-4764 15 28 series series NOUN american_scientific_journal-4764 15 29 of of ADP american_scientific_journal-4764 15 30 f f PROPN american_scientific_journal-4764 15 31 at at ADP american_scientific_journal-4764 15 32 x. x. PROPN american_scientific_journal-4764 15 33 ( ( PUNCT american_scientific_journal-4764 15 34 2.1 2.1 NUM american_scientific_journal-4764 15 35 ) ) PUNCT american_scientific_journal-4764 15 36 our our PRON american_scientific_journal-4764 15 37 theorem theorem ADJ american_scientific_journal-4764 15 38 fourier fourier PROPN american_scientific_journal-4764 15 39 series series PROPN american_scientific_journal-4764 15 40 by by ADP american_scientific_journal-4764 15 41 ( ( PUNCT american_scientific_journal-4764 15 42 n n CCONJ american_scientific_journal-4764 15 43 , , PUNCT american_scientific_journal-4764 15 44 p p X american_scientific_journal-4764 15 45 , , PUNCT american_scientific_journal-4764 15 46 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 15 47 ) ) PUNCT american_scientific_journal-4764 15 48 is be AUX american_scientific_journal-4764 15 49 the the DET american_scientific_journal-4764 15 50 analogues analogue NOUN american_scientific_journal-4764 15 51 to to ADP american_scientific_journal-4764 15 52 the the DET american_scientific_journal-4764 15 53 ( ( PUNCT american_scientific_journal-4764 15 54 e e NOUN american_scientific_journal-4764 15 55 , , PUNCT american_scientific_journal-4764 15 56 c c NOUN american_scientific_journal-4764 15 57 ) ) PUNCT american_scientific_journal-4764 15 58 means mean VERB american_scientific_journal-4764 15 59 theorem theorem VERB american_scientific_journal-4764 15 60 , , PUNCT american_scientific_journal-4764 15 61 which which PRON american_scientific_journal-4764 15 62 is be AUX american_scientific_journal-4764 15 63 as as SCONJ american_scientific_journal-4764 15 64 follows follow NOUN american_scientific_journal-4764 15 65 theorem theorem ADJ american_scientific_journal-4764 15 66 : : PUNCT american_scientific_journal-4764 15 67 let let VERB american_scientific_journal-4764 15 68 { { PUNCT american_scientific_journal-4764 15 69 𝐿𝐿𝑛𝑛 𝐿𝐿𝑛𝑛 VERB american_scientific_journal-4764 15 70 } } PUNCT american_scientific_journal-4764 15 71 and and CCONJ american_scientific_journal-4764 15 72 { { PUNCT american_scientific_journal-4764 15 73 π‘žπ‘žπ‘›π‘› π‘žπ‘žπ‘›π‘› NOUN american_scientific_journal-4764 15 74 } } PUNCT american_scientific_journal-4764 15 75 be be AUX american_scientific_journal-4764 15 76 the the DET american_scientific_journal-4764 15 77 positive positive ADJ american_scientific_journal-4764 15 78 monotonic monotonic NOUN american_scientific_journal-4764 15 79 , , PUNCT american_scientific_journal-4764 15 80 non non X american_scientific_journal-4764 15 81 increasing increase VERB american_scientific_journal-4764 15 82 sequence sequence NOUN american_scientific_journal-4764 15 83 of of ADP american_scientific_journal-4764 15 84 real real ADJ american_scientific_journal-4764 15 85 numbers number NOUN american_scientific_journal-4764 15 86 be be AUX american_scientific_journal-4764 15 87 summable summable ADJ american_scientific_journal-4764 15 88 ( ( PUNCT american_scientific_journal-4764 15 89 n n CCONJ american_scientific_journal-4764 15 90 , , PUNCT american_scientific_journal-4764 15 91 p p X american_scientific_journal-4764 15 92 , , PUNCT american_scientific_journal-4764 15 93 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 15 94 ) ) PUNCT american_scientific_journal-4764 15 95 to to ADP american_scientific_journal-4764 15 96 f(x f(x PROPN american_scientific_journal-4764 15 97 ) ) PUNCT american_scientific_journal-4764 15 98 at at ADP american_scientific_journal-4764 15 99 the the DET american_scientific_journal-4764 15 100 point point NOUN american_scientific_journal-4764 15 101 t t NOUN american_scientific_journal-4764 15 102 = = NOUN american_scientific_journal-4764 15 103 x x X american_scientific_journal-4764 15 104 is be AUX american_scientific_journal-4764 15 105 𝑑𝑑𝑁𝑁 𝑑𝑑𝑁𝑁 NOUN american_scientific_journal-4764 15 106 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 NUM american_scientific_journal-4764 15 107 βˆ’ βˆ’ PROPN american_scientific_journal-4764 15 108 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 15 109 ) ) PUNCT american_scientific_journal-4764 15 110 = = SYM american_scientific_journal-4764 15 111 π‘œπ‘œ(1 π‘œπ‘œ(1 NOUN american_scientific_journal-4764 15 112 ) ) PUNCT american_scientific_journal-4764 15 113 proof proof NOUN american_scientific_journal-4764 15 114 of of ADP american_scientific_journal-4764 15 115 the the DET american_scientific_journal-4764 15 116 above above ADJ american_scientific_journal-4764 15 117 theorem theorem NOUN american_scientific_journal-4764 15 118 required require VERB american_scientific_journal-4764 15 119 some some DET american_scientific_journal-4764 15 120 lemmas lemmas PROPN american_scientific_journal-4764 15 121 3 3 NUM american_scientific_journal-4764 15 122 . . PUNCT american_scientific_journal-4764 16 1 lemmas lemmas PROPN american_scientific_journal-4764 16 2 lemma lemma PROPN american_scientific_journal-4764 17 1 3.1for 3.1for NUM american_scientific_journal-4764 17 2 0 0 NUM american_scientific_journal-4764 17 3 ≀ ≀ NUM american_scientific_journal-4764 17 4 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 17 5 ≀ ≀ NUM american_scientific_journal-4764 17 6 1 1 NUM american_scientific_journal-4764 17 7 𝑛𝑛 𝑛𝑛 ADP american_scientific_journal-4764 17 8 |𝐾𝐾𝑛𝑛(𝑑𝑑)| |𝐾𝐾𝑛𝑛(𝑑𝑑)| NOUN american_scientific_journal-4764 17 9 = = SYM american_scientific_journal-4764 17 10 π‘œπ‘œ(𝑛𝑛 π‘œπ‘œ(𝑛𝑛 PROPN american_scientific_journal-4764 17 11 ) ) PUNCT american_scientific_journal-4764 18 1 lemma lemma PROPN american_scientific_journal-4764 19 1 3.2if 3.2if NUM american_scientific_journal-4764 19 2 { { PUNCT american_scientific_journal-4764 19 3 𝐿𝐿𝑛𝑛 𝐿𝐿𝑛𝑛 NOUN american_scientific_journal-4764 19 4 } } PUNCT american_scientific_journal-4764 19 5 and and CCONJ american_scientific_journal-4764 19 6 { { PUNCT american_scientific_journal-4764 19 7 π‘žπ‘žπ‘›π‘› π‘žπ‘žπ‘›π‘› NOUN american_scientific_journal-4764 19 8 } } PUNCT american_scientific_journal-4764 19 9 are be AUX american_scientific_journal-4764 19 10 non non X american_scientific_journal-4764 19 11 negative negative ADJ american_scientific_journal-4764 19 12 and and CCONJ american_scientific_journal-4764 19 13 non non ADJ american_scientific_journal-4764 19 14 increasing increase VERB american_scientific_journal-4764 19 15 , , PUNCT american_scientific_journal-4764 19 16 then then ADV american_scientific_journal-4764 19 17 for for ADP american_scientific_journal-4764 19 18 0 0 NUM american_scientific_journal-4764 19 19 ≀ ≀ NOUN american_scientific_journal-4764 19 20 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4764 19 21 ≀ ≀ NUM american_scientific_journal-4764 19 22 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4764 19 23 < < X american_scientific_journal-4764 19 24 ∞ ∞ PROPN american_scientific_journal-4764 19 25 , , PUNCT american_scientific_journal-4764 19 26 0 0 NUM american_scientific_journal-4764 19 27 ≀ ≀ NUM american_scientific_journal-4764 20 1 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 20 2 ≀ ≀ ADJ american_scientific_journal-4764 20 3 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4764 20 4 , , PUNCT american_scientific_journal-4764 20 5 and and CCONJ american_scientific_journal-4764 20 6 any any DET american_scientific_journal-4764 20 7 n n NOUN american_scientific_journal-4764 20 8 we we PRON american_scientific_journal-4764 20 9 have have VERB american_scientific_journal-4764 20 10 1 1 NUM american_scientific_journal-4764 20 11 2πœ‹πœ‹π‘…π‘…π‘›π‘› 2πœ‹πœ‹π‘…π‘…π‘›π‘› NUM american_scientific_journal-4764 20 12 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 13 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 20 14 πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿ πΏπΏπ‘›π‘›βˆ’π‘Ÿπ‘Ÿπ‘žπ‘žπ‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 20 15 π‘π‘π‘π‘π‘π‘π‘Ÿπ‘Ÿ π‘π‘π‘π‘π‘π‘π‘Ÿπ‘Ÿ PROPN american_scientific_journal-4764 20 16 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 17 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 20 18 2 2 NUM american_scientific_journal-4764 20 19 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 20 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 21 𝑐𝑐𝑠𝑠𝑛𝑛(π‘Ÿπ‘Ÿ+1) 𝑐𝑐𝑠𝑠𝑛𝑛(π‘Ÿπ‘Ÿ+1) PROPN american_scientific_journal-4764 20 22 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 23 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 20 24 2 2 NUM american_scientific_journal-4764 20 25 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 26 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 27 𝑐𝑐𝑠𝑠𝑛𝑛 𝑐𝑐𝑠𝑠𝑛𝑛 PROPN american_scientific_journal-4764 20 28 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 29 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 20 30 2 2 NUM american_scientific_journal-4764 20 31 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 32 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 33 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4764 20 34 π‘Ÿπ‘Ÿ=π‘Žπ‘Ž π‘Ÿπ‘Ÿ=π‘Žπ‘Ž NUM american_scientific_journal-4764 20 35 οΏ½ οΏ½ NOUN american_scientific_journal-4764 20 36 = = SYM american_scientific_journal-4764 20 37 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 20 38 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 39 π‘…π‘…π‘˜π‘˜ π‘…π‘…π‘˜π‘˜ PROPN american_scientific_journal-4764 20 40 𝑑𝑑𝑅𝑅𝑛𝑛 𝑑𝑑𝑅𝑅𝑛𝑛 NOUN american_scientific_journal-4764 20 41 οΏ½ οΏ½ PROPN american_scientific_journal-4764 20 42 4 4 NUM american_scientific_journal-4764 20 43 . . PUNCT american_scientific_journal-4764 21 1 proof proof NOUN american_scientific_journal-4764 21 2 of of ADP american_scientific_journal-4764 21 3 theorem theorem NOUN american_scientific_journal-4764 21 4 let let VERB american_scientific_journal-4764 21 5 f(t f(t NOUN american_scientific_journal-4764 21 6 ) ) PUNCT american_scientific_journal-4764 21 7 be be VERB american_scientific_journal-4764 21 8 a a DET american_scientific_journal-4764 21 9 periodic periodic ADJ american_scientific_journal-4764 21 10 function function NOUN american_scientific_journal-4764 21 11 with with ADP american_scientific_journal-4764 21 12 period period NOUN american_scientific_journal-4764 21 13 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4764 21 14 and and CCONJ american_scientific_journal-4764 21 15 integrable integrable ADJ american_scientific_journal-4764 21 16 in in ADP american_scientific_journal-4764 21 17 the the DET american_scientific_journal-4764 21 18 same same ADJ american_scientific_journal-4764 21 19 sense sense NOUN american_scientific_journal-4764 21 20 of of ADP american_scientific_journal-4764 21 21 lebesgue lebesgue NOUN american_scientific_journal-4764 21 22 over over ADP american_scientific_journal-4764 21 23 the the DET american_scientific_journal-4764 21 24 interval interval NOUN american_scientific_journal-4764 21 25 οΏ½ οΏ½ PROPN american_scientific_journal-4764 21 26 – – PUNCT american_scientific_journal-4764 21 27 πœ‹πœ‹,πœ‹πœ‹ πœ‹πœ‹,πœ‹πœ‹ PROPN american_scientific_journal-4764 21 28 οΏ½ οΏ½ PROPN american_scientific_journal-4764 21 29 let let VERB american_scientific_journal-4764 21 30 its its PRON american_scientific_journal-4764 21 31 fourier fourier NOUN american_scientific_journal-4764 21 32 series series NOUN american_scientific_journal-4764 21 33 be be AUX american_scientific_journal-4764 21 34 given give VERB american_scientific_journal-4764 21 35 by by ADP american_scientific_journal-4764 21 36 𝑓𝑓(𝑑𝑑)~ 𝑓𝑓(𝑑𝑑)~ PROPN american_scientific_journal-4764 21 37 1 1 NUM american_scientific_journal-4764 21 38 2 2 NUM american_scientific_journal-4764 21 39 π‘Žπ‘Ž0 π‘Žπ‘Ž0 NOUN american_scientific_journal-4764 22 1 + + CCONJ american_scientific_journal-4764 22 2 βˆ‘ βˆ‘ PROPN american_scientific_journal-4764 22 3 ( ( PUNCT american_scientific_journal-4764 22 4 π‘Žπ‘Žπ‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ π‘Žπ‘Žπ‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ NOUN american_scientific_journal-4764 22 5 + + CCONJ american_scientific_journal-4764 22 6 π‘π‘π‘›π‘›π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘›π‘›π‘‘π‘‘)∞ π‘π‘π‘›π‘›π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘›π‘›π‘‘π‘‘)∞ NOUN american_scientific_journal-4764 22 7 𝑛𝑛=1 𝑛𝑛=1 NOUN american_scientific_journal-4764 22 8 ( ( PUNCT american_scientific_journal-4764 22 9 4.1 4.1 NUM american_scientific_journal-4764 22 10 ) ) PUNCT american_scientific_journal-4764 22 11 following follow VERB american_scientific_journal-4764 22 12 zygmund zygmund PROPN american_scientific_journal-4764 22 13 [ [ X american_scientific_journal-4764 22 14 3 3 NUM american_scientific_journal-4764 22 15 ] ] PUNCT american_scientific_journal-4764 22 16 , , PUNCT american_scientific_journal-4764 22 17 the the DET american_scientific_journal-4764 22 18 nth nth NOUN american_scientific_journal-4764 22 19 sum sum NOUN american_scientific_journal-4764 22 20 π‘Žπ‘Žπ‘›π‘›(π‘₯π‘₯ π‘Žπ‘Žπ‘›π‘›(π‘₯π‘₯ PROPN american_scientific_journal-4764 22 21 ) ) PUNCT american_scientific_journal-4764 22 22 of of ADP american_scientific_journal-4764 22 23 the the DET american_scientific_journal-4764 22 24 series series NOUN american_scientific_journal-4764 22 25 at at ADP american_scientific_journal-4764 22 26 t t PROPN american_scientific_journal-4764 22 27 = = NOUN american_scientific_journal-4764 22 28 x x PROPN american_scientific_journal-4764 22 29 is be AUX american_scientific_journal-4764 22 30 given give VERB american_scientific_journal-4764 22 31 by by ADP american_scientific_journal-4764 22 32 american american ADJ american_scientific_journal-4764 22 33 scientific scientific ADJ american_scientific_journal-4764 22 34 research research PROPN american_scientific_journal-4764 22 35 journal journal NOUN american_scientific_journal-4764 22 36 for for ADP american_scientific_journal-4764 22 37 engineering engineering NOUN american_scientific_journal-4764 22 38 , , PUNCT american_scientific_journal-4764 22 39 technology technology NOUN american_scientific_journal-4764 22 40 , , PUNCT american_scientific_journal-4764 22 41 and and CCONJ american_scientific_journal-4764 22 42 sciences science NOUN american_scientific_journal-4764 22 43 ( ( PUNCT american_scientific_journal-4764 22 44 asrjets asrjet NOUN american_scientific_journal-4764 22 45 ) ) PUNCT american_scientific_journal-4764 22 46 ( ( PUNCT american_scientific_journal-4764 22 47 2019 2019 NUM american_scientific_journal-4764 22 48 ) ) PUNCT american_scientific_journal-4764 22 49 volume volume NOUN american_scientific_journal-4764 22 50 54 54 NUM american_scientific_journal-4764 22 51 , , PUNCT american_scientific_journal-4764 22 52 no no DET american_scientific_journal-4764 22 53 1 1 NUM american_scientific_journal-4764 22 54 , , PUNCT american_scientific_journal-4764 22 55 pp pp ADV american_scientific_journal-4764 22 56 131 131 NUM american_scientific_journal-4764 22 57 - - SYM american_scientific_journal-4764 22 58 136 136 NUM american_scientific_journal-4764 22 59 134 134 NUM american_scientific_journal-4764 22 60 π‘Žπ‘Žπ‘›π‘›(π‘₯π‘₯ π‘Žπ‘Žπ‘›π‘›(π‘₯π‘₯ NOUN american_scientific_journal-4764 22 61 ) ) PUNCT american_scientific_journal-4764 22 62 = = SYM american_scientific_journal-4764 22 63 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 22 64 ) ) PUNCT american_scientific_journal-4764 23 1 + + CCONJ american_scientific_journal-4764 23 2 1 1 NUM american_scientific_journal-4764 23 3 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4764 23 4 ∫ ∫ PROPN american_scientific_journal-4764 24 1 βˆ…π‘₯π‘₯(𝑑𝑑)πœ‹πœ‹ βˆ…π‘₯π‘₯(𝑑𝑑)πœ‹πœ‹ PROPN american_scientific_journal-4764 24 2 0 0 NUM american_scientific_journal-4764 24 3 𝑐𝑐𝑠𝑠𝑛𝑛(𝑛𝑛+1)𝑑𝑑 𝑐𝑐𝑠𝑠𝑛𝑛(𝑛𝑛+1)𝑑𝑑 NOUN american_scientific_journal-4764 24 4 𝑐𝑐𝑠𝑠𝑛𝑛 𝑐𝑐𝑠𝑠𝑛𝑛 NOUN american_scientific_journal-4764 24 5 οΏ½ οΏ½ PROPN american_scientific_journal-4764 24 6 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 24 7 2 2 NUM american_scientific_journal-4764 24 8 οΏ½ οΏ½ PROPN american_scientific_journal-4764 24 9 οΏ½ οΏ½ PROPN american_scientific_journal-4764 24 10 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 24 11 ( ( PUNCT american_scientific_journal-4764 24 12 4.2 4.2 NUM american_scientific_journal-4764 24 13 ) ) PUNCT american_scientific_journal-4764 25 1 so so SCONJ american_scientific_journal-4764 25 2 the the DET american_scientific_journal-4764 25 3 ( ( PUNCT american_scientific_journal-4764 25 4 e,1 e,1 NOUN american_scientific_journal-4764 25 5 ) ) PUNCT american_scientific_journal-4764 25 6 mean mean NOUN american_scientific_journal-4764 25 7 of of ADP american_scientific_journal-4764 25 8 the the DET american_scientific_journal-4764 25 9 series series NOUN american_scientific_journal-4764 25 10 at at ADP american_scientific_journal-4764 25 11 t t PROPN american_scientific_journal-4764 25 12 = = NOUN american_scientific_journal-4764 25 13 x x PROPN american_scientific_journal-4764 25 14 is be AUX american_scientific_journal-4764 25 15 given give VERB american_scientific_journal-4764 25 16 by by ADP american_scientific_journal-4764 25 17 𝐸𝐸𝑛𝑛1(π‘₯π‘₯ 𝐸𝐸𝑛𝑛1(π‘₯π‘₯ PROPN american_scientific_journal-4764 25 18 ) ) PUNCT american_scientific_journal-4764 26 1 = = SYM american_scientific_journal-4764 26 2 1 1 NUM american_scientific_journal-4764 26 3 2𝑛𝑛 2𝑛𝑛 NOUN american_scientific_journal-4764 26 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 26 5 οΏ½ οΏ½ PROPN american_scientific_journal-4764 26 6 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 26 7 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4764 26 8 οΏ½ οΏ½ PROPN american_scientific_journal-4764 26 9 𝑛𝑛 𝑛𝑛 VERB american_scientific_journal-4764 26 10 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 26 11 π‘Žπ‘Žπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘Žπ‘Žπ‘Ÿπ‘Ÿ(π‘₯π‘₯ PROPN american_scientific_journal-4764 26 12 ) ) PUNCT american_scientific_journal-4764 26 13 = = SYM american_scientific_journal-4764 26 14 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 26 15 ) ) PUNCT american_scientific_journal-4764 27 1 + + CCONJ american_scientific_journal-4764 27 2 1 1 NUM american_scientific_journal-4764 27 3 2𝑛𝑛+1πœ‹πœ‹ 2𝑛𝑛+1πœ‹πœ‹ ADJ american_scientific_journal-4764 27 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 27 6 ) ) PUNCT american_scientific_journal-4764 27 7 π‘Žπ‘ŽπΏπΏπ‘›π‘› π‘Žπ‘ŽπΏπΏπ‘›π‘› PROPN american_scientific_journal-4764 27 8 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 9 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 27 10 2 2 NUM american_scientific_journal-4764 27 11 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 12 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 13 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 27 14 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4764 27 15 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 16 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 17 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 18 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 27 19 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4764 27 20 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 21 𝑛𝑛 𝑛𝑛 VERB american_scientific_journal-4764 27 22 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PRON american_scientific_journal-4764 27 23 π‘Žπ‘ŽπΏπΏπ‘›π‘› π‘Žπ‘ŽπΏπΏπ‘›π‘› PROPN american_scientific_journal-4764 27 24 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 25 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4764 27 26 + + NOUN american_scientific_journal-4764 27 27 1 1 NUM american_scientific_journal-4764 27 28 2 2 NUM american_scientific_journal-4764 27 29 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 30 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 27 31 οΏ½ οΏ½ PROPN american_scientific_journal-4764 27 32 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 27 33 = = SYM american_scientific_journal-4764 27 34 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 27 35 ) ) PUNCT american_scientific_journal-4764 28 1 + + CCONJ american_scientific_journal-4764 28 2 1 1 NUM american_scientific_journal-4764 28 3 2𝑛𝑛+1πœ‹πœ‹ 2𝑛𝑛+1πœ‹πœ‹ ADJ american_scientific_journal-4764 28 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 28 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 28 6 ) ) PUNCT american_scientific_journal-4764 28 7 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 PROPN american_scientific_journal-4764 28 8 ) ) PUNCT american_scientific_journal-4764 28 9 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 NOUN american_scientific_journal-4764 28 10 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 28 11 0 0 NUM american_scientific_journal-4764 28 12 οΏ½ οΏ½ PROPN american_scientific_journal-4764 28 13 𝑒𝑒𝑠𝑠𝑑𝑑/2(1 𝑒𝑒𝑠𝑠𝑑𝑑/2(1 PROPN american_scientific_journal-4764 28 14 + + SYM american_scientific_journal-4764 28 15 𝑒𝑒𝑠𝑠𝑑𝑑)𝑛𝑛 𝑒𝑒𝑠𝑠𝑑𝑑)𝑛𝑛 NOUN american_scientific_journal-4764 28 16 οΏ½ οΏ½ NOUN american_scientific_journal-4764 28 17 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 28 18 = = SYM american_scientific_journal-4764 28 19 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 28 20 ) ) PUNCT american_scientific_journal-4764 29 1 + + CCONJ american_scientific_journal-4764 29 2 1 1 NUM american_scientific_journal-4764 29 3 2𝑛𝑛+1πœ‹πœ‹ 2𝑛𝑛+1πœ‹πœ‹ NUM american_scientific_journal-4764 29 4 ∫ ∫ PROPN american_scientific_journal-4764 29 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 29 6 ) ) PUNCT american_scientific_journal-4764 29 7 𝑐𝑐𝑠𝑠𝑛𝑛(𝑑𝑑/2 𝑐𝑐𝑠𝑠𝑛𝑛(𝑑𝑑/2 PART american_scientific_journal-4764 29 8 ) ) PUNCT american_scientific_journal-4764 29 9 πΌπΌπΌπΌπœ‹πœ‹ πΌπΌπΌπΌπœ‹πœ‹ PROPN american_scientific_journal-4764 29 10 0 0 NUM american_scientific_journal-4764 29 11 οΏ½ οΏ½ PROPN american_scientific_journal-4764 29 12 𝑒𝑒𝑠𝑠𝑑𝑑/2(1 𝑒𝑒𝑠𝑠𝑑𝑑/2(1 NOUN american_scientific_journal-4764 29 13 + + CCONJ american_scientific_journal-4764 29 14 π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘‘π‘‘ π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘‘π‘‘ ADJ american_scientific_journal-4764 29 15 + + X american_scientific_journal-4764 29 16 πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘›π‘‘π‘‘)𝑛𝑛 πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘›π‘‘π‘‘)𝑛𝑛 NOUN american_scientific_journal-4764 29 17 οΏ½ οΏ½ NOUN american_scientific_journal-4764 29 18 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 29 19 ( ( PUNCT american_scientific_journal-4764 29 20 4.3 4.3 NUM american_scientific_journal-4764 29 21 ) ) PUNCT american_scientific_journal-4764 29 22 = = SYM american_scientific_journal-4764 29 23 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 29 24 ) ) PUNCT american_scientific_journal-4764 30 1 + + CCONJ american_scientific_journal-4764 30 2 1 1 NUM american_scientific_journal-4764 30 3 2𝑛𝑛+1πœ‹πœ‹ 2𝑛𝑛+1πœ‹πœ‹ ADJ american_scientific_journal-4764 30 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 30 6 ) ) PUNCT american_scientific_journal-4764 30 7 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 PROPN american_scientific_journal-4764 30 8 ) ) PUNCT american_scientific_journal-4764 30 9 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 NOUN american_scientific_journal-4764 30 10 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 30 11 0 0 NUM american_scientific_journal-4764 30 12 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 13 𝑒𝑒𝑠𝑠𝑑𝑑/22π‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘› 𝑒𝑒𝑠𝑠𝑑𝑑/22π‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘› PROPN american_scientific_journal-4764 30 14 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 15 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 30 16 2 2 NUM american_scientific_journal-4764 30 17 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 18 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 19 π‘π‘π‘œπ‘œπ‘Žπ‘Ž π‘π‘π‘œπ‘œπ‘Žπ‘Ž NOUN american_scientific_journal-4764 30 20 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 30 21 2 2 NUM american_scientific_journal-4764 30 22 + + CCONJ american_scientific_journal-4764 30 23 πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘› πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘› NOUN american_scientific_journal-4764 30 24 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 30 25 2 2 NUM american_scientific_journal-4764 30 26 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 27 𝑛𝑛 𝑛𝑛 ADP american_scientific_journal-4764 30 28 οΏ½ οΏ½ PROPN american_scientific_journal-4764 30 29 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 30 30 = = PROPN american_scientific_journal-4764 30 31 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 30 32 ) ) PUNCT american_scientific_journal-4764 31 1 + + CCONJ american_scientific_journal-4764 31 2 1 1 NUM american_scientific_journal-4764 31 3 2𝑛𝑛+1πœ‹πœ‹ 2𝑛𝑛+1πœ‹πœ‹ ADJ american_scientific_journal-4764 31 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 31 6 ) ) PUNCT american_scientific_journal-4764 31 7 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑑𝑑/2 PROPN american_scientific_journal-4764 31 8 ) ) PUNCT american_scientific_journal-4764 31 9 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 NOUN american_scientific_journal-4764 31 10 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 31 11 0 0 NUM american_scientific_journal-4764 31 12 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 13 𝑒𝑒𝑠𝑠𝑑𝑑/22π‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘› 𝑒𝑒𝑠𝑠𝑑𝑑/22π‘›π‘›π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘› PROPN american_scientific_journal-4764 31 14 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 15 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 31 16 2 2 NUM american_scientific_journal-4764 31 17 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 18 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 19 π‘π‘π‘œπ‘œπ‘Žπ‘Ž π‘π‘π‘œπ‘œπ‘Žπ‘Ž NOUN american_scientific_journal-4764 31 20 𝑛𝑛𝑑𝑑 𝑛𝑛𝑑𝑑 NOUN american_scientific_journal-4764 31 21 2 2 NUM american_scientific_journal-4764 31 22 + + CCONJ american_scientific_journal-4764 31 23 πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘› πΏπΏπ‘Žπ‘ŽπΏπΏπ‘›π‘› ADV american_scientific_journal-4764 31 24 𝑛𝑛𝑑𝑑 𝑛𝑛𝑑𝑑 ADJ american_scientific_journal-4764 31 25 2 2 NUM american_scientific_journal-4764 31 26 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 27 οΏ½ οΏ½ PROPN american_scientific_journal-4764 31 28 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4764 31 29 = = PROPN american_scientific_journal-4764 31 30 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 31 31 ) ) PUNCT american_scientific_journal-4764 32 1 + + CCONJ american_scientific_journal-4764 32 2 1 1 NUM american_scientific_journal-4764 32 3 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4764 32 4 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 5 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 32 6 ) ) PUNCT american_scientific_journal-4764 32 7 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 32 8 0 0 NUM american_scientific_journal-4764 32 9 π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘›(𝑑𝑑/2)π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑛𝑛 π‘π‘π‘œπ‘œπ‘Žπ‘Žπ‘›π‘›(𝑑𝑑/2)π‘Žπ‘ŽπΏπΏπ‘›π‘›(𝑛𝑛 NOUN american_scientific_journal-4764 32 10 + + CCONJ american_scientific_journal-4764 32 11 1)(𝑑𝑑/2 1)(𝑑𝑑/2 NUM american_scientific_journal-4764 32 12 ) ) PUNCT american_scientific_journal-4764 32 13 sin sin NOUN american_scientific_journal-4764 32 14 ( ( PUNCT american_scientific_journal-4764 32 15 𝑑𝑑 𝑑𝑑 NOUN american_scientific_journal-4764 32 16 2 2 X american_scientific_journal-4764 32 17 ) ) PUNCT american_scientific_journal-4764 32 18 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 32 19 therefore therefore ADV american_scientific_journal-4764 32 20 𝑑𝑑𝑛𝑛 𝑑𝑑𝑛𝑛 PROPN american_scientific_journal-4764 32 21 𝑝𝑝,π‘žπ‘ž,𝐸𝐸(π‘₯π‘₯ 𝑝𝑝,π‘žπ‘ž,𝐸𝐸(π‘₯π‘₯ PROPN american_scientific_journal-4764 32 22 ) ) PUNCT american_scientific_journal-4764 32 23 βˆ’ βˆ’ PROPN american_scientific_journal-4764 32 24 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 32 25 ) ) PUNCT american_scientific_journal-4764 32 26 = = SYM american_scientific_journal-4764 32 27 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 28 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 29 + + CCONJ american_scientific_journal-4764 32 30 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 31 + + NOUN american_scientific_journal-4764 32 32 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 33 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 32 34 𝛿𝛿 𝛿𝛿 PROPN american_scientific_journal-4764 32 35 𝛿𝛿 𝛿𝛿 PROPN american_scientific_journal-4764 32 36 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 32 37 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 32 38 0 0 NUM american_scientific_journal-4764 32 39 οΏ½ οΏ½ PROPN american_scientific_journal-4764 32 40 𝐾𝐾𝑛𝑛(𝑑𝑑)βˆ…π‘₯π‘₯(𝑑𝑑)𝑑𝑑𝑑𝑑 𝐾𝐾𝑛𝑛(𝑑𝑑)βˆ…π‘₯π‘₯(𝑑𝑑)𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4764 32 41 = = SYM american_scientific_journal-4764 32 42 𝐼𝐼1 𝐼𝐼1 PROPN american_scientific_journal-4764 33 1 + + CCONJ american_scientific_journal-4764 33 2 𝐼𝐼2 𝐼𝐼2 PROPN american_scientific_journal-4764 33 3 + + CCONJ american_scientific_journal-4764 33 4 𝐼𝐼3 𝐼𝐼3 PROPN american_scientific_journal-4764 33 5 ( ( PUNCT american_scientific_journal-4764 33 6 say say INTJ american_scientific_journal-4764 33 7 ) ) PUNCT american_scientific_journal-4764 33 8 ( ( PUNCT american_scientific_journal-4764 33 9 4.4 4.4 NUM american_scientific_journal-4764 33 10 ) ) PUNCT american_scientific_journal-4764 33 11 we we PRON american_scientific_journal-4764 33 12 have have VERB american_scientific_journal-4764 33 13 |𝐼𝐼1| |𝐼𝐼1| PROPN american_scientific_journal-4764 33 14 ≀ ≀ ADJ american_scientific_journal-4764 33 15 οΏ½ οΏ½ PROPN american_scientific_journal-4764 33 16 |𝐾𝐾𝑛𝑛(𝑑𝑑)| |𝐾𝐾𝑛𝑛(𝑑𝑑)| PROPN american_scientific_journal-4764 33 17 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 33 18 0 0 NUM american_scientific_journal-4764 34 1 |βˆ…π‘₯π‘₯(𝑑𝑑)|𝑑𝑑𝑑𝑑 |βˆ…π‘₯π‘₯(𝑑𝑑)|𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 34 2 = = SYM american_scientific_journal-4764 34 3 𝑂𝑂(𝑛𝑛)∫ 𝑂𝑂(𝑛𝑛)∫ PROPN american_scientific_journal-4764 34 4 |βˆ…π‘₯π‘₯(𝑑𝑑)|1/𝑛𝑛 |βˆ…π‘₯π‘₯(𝑑𝑑)|1/𝑛𝑛 NOUN american_scientific_journal-4764 34 5 0 0 NUM american_scientific_journal-4764 34 6 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4764 34 7 ( ( PUNCT american_scientific_journal-4764 34 8 π‘’π‘’π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘’π‘’ π‘’π‘’π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘’π‘’ PROPN american_scientific_journal-4764 34 9 πΏπΏπ‘’π‘’πΌπΌπΌπΌπ‘Žπ‘Ž πΏπΏπ‘’π‘’πΌπΌπΌπΌπ‘Žπ‘Ž PROPN american_scientific_journal-4764 34 10 3.1 3.1 NUM american_scientific_journal-4764 34 11 ) ) PUNCT american_scientific_journal-4764 34 12 ( ( PUNCT american_scientific_journal-4764 34 13 4.5 4.5 NUM american_scientific_journal-4764 34 14 ) ) PUNCT american_scientific_journal-4764 34 15 american american ADJ american_scientific_journal-4764 34 16 scientific scientific ADJ american_scientific_journal-4764 34 17 research research NOUN american_scientific_journal-4764 34 18 journal journal NOUN american_scientific_journal-4764 34 19 for for ADP american_scientific_journal-4764 34 20 engineering engineering NOUN american_scientific_journal-4764 34 21 , , PUNCT american_scientific_journal-4764 34 22 technology technology NOUN american_scientific_journal-4764 34 23 , , PUNCT american_scientific_journal-4764 34 24 and and CCONJ american_scientific_journal-4764 34 25 sciences science NOUN american_scientific_journal-4764 34 26 ( ( PUNCT american_scientific_journal-4764 34 27 asrjets asrjet NOUN american_scientific_journal-4764 34 28 ) ) PUNCT american_scientific_journal-4764 34 29 ( ( PUNCT american_scientific_journal-4764 34 30 2019 2019 NUM american_scientific_journal-4764 34 31 ) ) PUNCT american_scientific_journal-4764 34 32 volume volume NOUN american_scientific_journal-4764 34 33 54 54 NUM american_scientific_journal-4764 34 34 , , PUNCT american_scientific_journal-4764 34 35 no no DET american_scientific_journal-4764 34 36 1 1 NUM american_scientific_journal-4764 34 37 , , PUNCT american_scientific_journal-4764 34 38 pp pp ADV american_scientific_journal-4764 34 39 131 131 NUM american_scientific_journal-4764 34 40 - - SYM american_scientific_journal-4764 34 41 136 136 NUM american_scientific_journal-4764 34 42 135 135 NUM american_scientific_journal-4764 34 43 = = SYM american_scientific_journal-4764 34 44 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 45 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 46 1 1 NUM american_scientific_journal-4764 34 47 𝛼𝛼(𝑛𝑛 𝛼𝛼(𝑛𝑛 PROPN american_scientific_journal-4764 34 48 ) ) PUNCT american_scientific_journal-4764 34 49 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 50 = = SYM american_scientific_journal-4764 34 51 π‘œπ‘œ(1 π‘œπ‘œ(1 PROPN american_scientific_journal-4764 34 52 ) ) PUNCT american_scientific_journal-4764 34 53 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 34 54 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 34 55 β†’ β†’ PUNCT american_scientific_journal-4764 34 56 ∞ ∞ PROPN american_scientific_journal-4764 34 57 ( ( PUNCT american_scientific_journal-4764 34 58 4.6 4.6 NUM american_scientific_journal-4764 34 59 ) ) PUNCT american_scientific_journal-4764 34 60 now now ADV american_scientific_journal-4764 34 61 |𝐼𝐼2| |𝐼𝐼2| ADV american_scientific_journal-4764 34 62 ≀ ≀ PROPN american_scientific_journal-4764 34 63 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 64 |𝐾𝐾𝑛𝑛(𝑑𝑑)| |𝐾𝐾𝑛𝑛(𝑑𝑑)| PROPN american_scientific_journal-4764 34 65 𝛿𝛿 𝛿𝛿 NOUN american_scientific_journal-4764 34 66 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 67 |βˆ…π‘₯π‘₯(π‘₯π‘₯)|𝑑𝑑𝑑𝑑 |βˆ…π‘₯π‘₯(π‘₯π‘₯)|𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4764 34 68 ( ( PUNCT american_scientific_journal-4764 34 69 π‘€π‘€β„Žπ‘’π‘’π‘Ÿπ‘Ÿπ‘’π‘’ π‘€π‘€β„Žπ‘’π‘’π‘Ÿπ‘Ÿπ‘’π‘’ NOUN american_scientific_journal-4764 34 70 0 0 PUNCT american_scientific_journal-4764 34 71 < < X american_scientific_journal-4764 34 72 𝛿𝛿 𝛿𝛿 X american_scientific_journal-4764 34 73 < < X american_scientific_journal-4764 34 74 1 1 NUM american_scientific_journal-4764 34 75 ) ) PUNCT american_scientific_journal-4764 34 76 = = SYM american_scientific_journal-4764 34 77 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 78 π‘œπ‘œ π‘œπ‘œ PROPN american_scientific_journal-4764 34 79 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 80 𝑅𝑅(1/𝑑𝑑 𝑅𝑅(1/𝑑𝑑 NOUN american_scientific_journal-4764 34 81 ) ) PUNCT american_scientific_journal-4764 34 82 𝑑𝑑𝑅𝑅(𝑛𝑛 𝑑𝑑𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 83 ) ) PUNCT american_scientific_journal-4764 34 84 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 85 |βˆ…π‘₯π‘₯(𝑑𝑑)| |βˆ…π‘₯π‘₯(𝑑𝑑)| PROPN american_scientific_journal-4764 34 86 𝛿𝛿 𝛿𝛿 NOUN american_scientific_journal-4764 34 87 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 88 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4764 34 89 ( ( PUNCT american_scientific_journal-4764 34 90 π‘’π‘’π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘’π‘’ π‘’π‘’π‘Žπ‘ŽπΏπΏπ‘›π‘›π‘’π‘’ PROPN american_scientific_journal-4764 34 91 πΏπΏπ‘’π‘’πΌπΌπΌπΌπ‘Žπ‘Ž πΏπΏπ‘’π‘’πΌπΌπΌπΌπ‘Žπ‘Ž PROPN american_scientific_journal-4764 34 92 3.2 3.2 NUM american_scientific_journal-4764 34 93 ) ) PUNCT american_scientific_journal-4764 34 94 = = SYM american_scientific_journal-4764 34 95 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 96 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 97 1 1 NUM american_scientific_journal-4764 34 98 𝑅𝑅(𝑛𝑛 𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 99 ) ) PUNCT american_scientific_journal-4764 34 100 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 101 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 102 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 103 𝑅𝑅(1/𝑑𝑑 𝑅𝑅(1/𝑑𝑑 NOUN american_scientific_journal-4764 34 104 ) ) PUNCT american_scientific_journal-4764 34 105 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4764 34 106 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 107 |βˆ…π‘₯π‘₯(𝑑𝑑)| |βˆ…π‘₯π‘₯(𝑑𝑑)| NOUN american_scientific_journal-4764 34 108 𝛿𝛿 𝛿𝛿 NOUN american_scientific_journal-4764 34 109 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 110 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4764 34 111 = = SYM american_scientific_journal-4764 34 112 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 113 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 114 1 1 NUM american_scientific_journal-4764 34 115 𝑅𝑅(𝑛𝑛 𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 116 ) ) PUNCT american_scientific_journal-4764 34 117 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 118 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 119 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 120 𝑅𝑅(1/𝑑𝑑 𝑅𝑅(1/𝑑𝑑 NOUN american_scientific_journal-4764 34 121 ) ) PUNCT american_scientific_journal-4764 34 122 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 34 123 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 34 124 ) ) PUNCT american_scientific_journal-4764 34 125 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 126 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 127 𝛿𝛿 𝛿𝛿 PROPN american_scientific_journal-4764 34 128 βˆ’ βˆ’ PROPN american_scientific_journal-4764 34 129 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 130 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 34 131 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 132 𝑅𝑅(1/𝑑𝑑 𝑅𝑅(1/𝑑𝑑 NOUN american_scientific_journal-4764 34 133 ) ) PUNCT american_scientific_journal-4764 34 134 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 34 135 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 136 𝛿𝛿 𝛿𝛿 PROPN american_scientific_journal-4764 34 137 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 138 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 34 139 ) ) PUNCT american_scientific_journal-4764 34 140 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 141 = = SYM american_scientific_journal-4764 34 142 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 143 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 144 1 1 NUM american_scientific_journal-4764 34 145 𝑅𝑅(𝑛𝑛 𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 146 ) ) PUNCT american_scientific_journal-4764 34 147 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 148 + + CCONJ american_scientific_journal-4764 34 149 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 150 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 151 1 1 NUM american_scientific_journal-4764 34 152 𝛼𝛼(𝑛𝑛 𝛼𝛼(𝑛𝑛 PROPN american_scientific_journal-4764 34 153 ) ) PUNCT american_scientific_journal-4764 34 154 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 155 + + CCONJ american_scientific_journal-4764 34 156 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 157 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 158 1 1 NUM american_scientific_journal-4764 34 159 𝑅𝑅(𝑛𝑛 𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 160 ) ) PUNCT american_scientific_journal-4764 34 161 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 162 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 163 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 164 βˆ…π‘₯π‘₯(𝑑𝑑 βˆ…π‘₯π‘₯(𝑑𝑑 PROPN american_scientific_journal-4764 34 165 ) ) PUNCT american_scientific_journal-4764 34 166 𝛿𝛿 𝛿𝛿 NOUN american_scientific_journal-4764 34 167 1/𝑛𝑛 1/𝑛𝑛 NUM american_scientific_journal-4764 34 168 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 169 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4764 34 170 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 171 𝑅𝑅(1/𝑑𝑑)𝛼𝛼(1/𝑑𝑑 𝑅𝑅(1/𝑑𝑑)𝛼𝛼(1/𝑑𝑑 NOUN american_scientific_journal-4764 34 172 ) ) PUNCT american_scientific_journal-4764 34 173 𝑑𝑑𝛼𝛼(1/𝑑𝑑 𝑑𝑑𝛼𝛼(1/𝑑𝑑 PROPN american_scientific_journal-4764 34 174 ) ) PUNCT american_scientific_journal-4764 34 175 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 176 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 177 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 178 = = SYM american_scientific_journal-4764 34 179 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 180 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 181 1 1 NUM american_scientific_journal-4764 34 182 𝑅𝑅(𝑛𝑛 𝑅𝑅(𝑛𝑛 PROPN american_scientific_journal-4764 34 183 ) ) PUNCT american_scientific_journal-4764 34 184 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 185 + + CCONJ american_scientific_journal-4764 34 186 π‘œπ‘œ π‘œπ‘œ PRON american_scientific_journal-4764 34 187 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 188 1 1 NUM american_scientific_journal-4764 34 189 𝛼𝛼(𝑛𝑛 𝛼𝛼(𝑛𝑛 PROPN american_scientific_journal-4764 34 190 ) ) PUNCT american_scientific_journal-4764 34 191 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 192 + + CCONJ american_scientific_journal-4764 34 193 π‘œπ‘œ(1 π‘œπ‘œ(1 PROPN american_scientific_journal-4764 34 194 ) ) PUNCT american_scientific_journal-4764 34 195 = = SYM american_scientific_journal-4764 34 196 π‘œπ‘œ(1 π‘œπ‘œ(1 NOUN american_scientific_journal-4764 34 197 ) ) PUNCT american_scientific_journal-4764 34 198 , , PUNCT american_scientific_journal-4764 34 199 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 34 200 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 34 201 β†’ β†’ PUNCT american_scientific_journal-4764 34 202 ∞ ∞ PROPN american_scientific_journal-4764 34 203 ( ( PUNCT american_scientific_journal-4764 34 204 4.7 4.7 NUM american_scientific_journal-4764 34 205 ) ) PUNCT american_scientific_journal-4764 34 206 now now ADV american_scientific_journal-4764 34 207 𝐼𝐼3 𝐼𝐼3 PROPN american_scientific_journal-4764 34 208 = = SYM american_scientific_journal-4764 34 209 οΏ½ οΏ½ PROPN american_scientific_journal-4764 34 210 |𝐾𝐾𝑛𝑛(𝑑𝑑)||βˆ…π‘₯π‘₯(𝑑𝑑)| |𝐾𝐾𝑛𝑛(𝑑𝑑)||βˆ…π‘₯π‘₯(𝑑𝑑)| PROPN american_scientific_journal-4764 34 211 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4764 34 212 𝛿𝛿 𝛿𝛿 NOUN american_scientific_journal-4764 34 213 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4764 34 214 by by ADP american_scientific_journal-4764 34 215 riemann riemann PROPN american_scientific_journal-4764 34 216 - - PUNCT american_scientific_journal-4764 34 217 lebesgue lebesgue NOUN american_scientific_journal-4764 34 218 theorem theorem NOUN american_scientific_journal-4764 34 219 and and CCONJ american_scientific_journal-4764 34 220 regularity regularity NOUN american_scientific_journal-4764 34 221 of of ADP american_scientific_journal-4764 34 222 the the DET american_scientific_journal-4764 34 223 method method NOUN american_scientific_journal-4764 34 224 of of ADP american_scientific_journal-4764 34 225 summability summability NOUN american_scientific_journal-4764 34 226 we we PRON american_scientific_journal-4764 34 227 have have AUX american_scientific_journal-4764 34 228 𝐼𝐼3 𝐼𝐼3 PROPN american_scientific_journal-4764 34 229 = = PUNCT american_scientific_journal-4764 34 230 π‘œπ‘œ(1 π‘œπ‘œ(1 PROPN american_scientific_journal-4764 34 231 ) ) PUNCT american_scientific_journal-4764 34 232 , , PUNCT american_scientific_journal-4764 34 233 π‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4764 34 234 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4764 34 235 β†’ β†’ PUNCT american_scientific_journal-4764 34 236 ∞ ∞ PROPN american_scientific_journal-4764 34 237 ( ( PUNCT american_scientific_journal-4764 34 238 4.8 4.8 NUM american_scientific_journal-4764 34 239 ) ) PUNCT american_scientific_journal-4764 34 240 combining combine VERB american_scientific_journal-4764 34 241 ( ( PUNCT american_scientific_journal-4764 34 242 4.6),(4.7 4.6),(4.7 NUM american_scientific_journal-4764 34 243 ) ) PUNCT american_scientific_journal-4764 34 244 and and CCONJ american_scientific_journal-4764 34 245 ( ( PUNCT american_scientific_journal-4764 34 246 4.8 4.8 NUM american_scientific_journal-4764 34 247 ) ) PUNCT american_scientific_journal-4764 34 248 we we PRON american_scientific_journal-4764 34 249 get get VERB american_scientific_journal-4764 34 250 𝑑𝑑𝑁𝑁 𝑑𝑑𝑁𝑁 NOUN american_scientific_journal-4764 34 251 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 𝑝𝑝,π‘žπ‘ž,𝐸𝐸 PROPN american_scientific_journal-4764 34 252 βˆ’ βˆ’ PROPN american_scientific_journal-4764 34 253 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-4764 34 254 ) ) PUNCT american_scientific_journal-4764 35 1 = = SYM american_scientific_journal-4764 35 2 π‘œπ‘œ(1 π‘œπ‘œ(1 PROPN american_scientific_journal-4764 35 3 ) ) PUNCT american_scientific_journal-4764 35 4 this this PRON american_scientific_journal-4764 35 5 completes complete VERB american_scientific_journal-4764 35 6 the the DET american_scientific_journal-4764 35 7 proof proof NOUN american_scientific_journal-4764 35 8 of of ADP american_scientific_journal-4764 35 9 the the DET american_scientific_journal-4764 35 10 theorem theorem PROPN american_scientific_journal-4764 35 11 . . PUNCT american_scientific_journal-4764 36 1 american american PROPN american_scientific_journal-4764 36 2 scientific scientific PROPN american_scientific_journal-4764 36 3 research research PROPN american_scientific_journal-4764 36 4 journal journal NOUN american_scientific_journal-4764 36 5 for for ADP american_scientific_journal-4764 36 6 engineering engineering NOUN american_scientific_journal-4764 36 7 , , PUNCT american_scientific_journal-4764 36 8 technology technology NOUN american_scientific_journal-4764 36 9 , , PUNCT american_scientific_journal-4764 36 10 and and CCONJ american_scientific_journal-4764 36 11 sciences science NOUN american_scientific_journal-4764 36 12 ( ( PUNCT american_scientific_journal-4764 36 13 asrjets asrjet NOUN american_scientific_journal-4764 36 14 ) ) PUNCT american_scientific_journal-4764 36 15 ( ( PUNCT american_scientific_journal-4764 36 16 2019 2019 NUM american_scientific_journal-4764 36 17 ) ) PUNCT american_scientific_journal-4764 36 18 volume volume NOUN american_scientific_journal-4764 36 19 54 54 NUM american_scientific_journal-4764 36 20 , , PUNCT american_scientific_journal-4764 36 21 no no DET american_scientific_journal-4764 36 22 1 1 NUM american_scientific_journal-4764 36 23 , , PUNCT american_scientific_journal-4764 36 24 pp pp ADV american_scientific_journal-4764 36 25 131 131 NUM american_scientific_journal-4764 36 26 - - SYM american_scientific_journal-4764 36 27 136 136 NUM american_scientific_journal-4764 36 28 136 136 NUM american_scientific_journal-4764 36 29 5 5 NUM american_scientific_journal-4764 36 30 . . PUNCT american_scientific_journal-4764 37 1 conclusion conclusion NOUN american_scientific_journal-4764 37 2 we we PRON american_scientific_journal-4764 37 3 conclude conclude VERB american_scientific_journal-4764 37 4 that that SCONJ american_scientific_journal-4764 37 5 the the DET american_scientific_journal-4764 37 6 above above ADJ american_scientific_journal-4764 37 7 theorem theorem NOUN american_scientific_journal-4764 37 8 which which PRON american_scientific_journal-4764 37 9 is be AUX american_scientific_journal-4764 37 10 proved prove VERB american_scientific_journal-4764 37 11 in in ADP american_scientific_journal-4764 37 12 ( ( PUNCT american_scientific_journal-4764 37 13 e e NOUN american_scientific_journal-4764 37 14 , , PUNCT american_scientific_journal-4764 37 15 c c NOUN american_scientific_journal-4764 37 16 ) ) PUNCT american_scientific_journal-4764 37 17 means mean NOUN american_scientific_journal-4764 37 18 can can AUX american_scientific_journal-4764 37 19 be be AUX american_scientific_journal-4764 37 20 proved prove VERB american_scientific_journal-4764 37 21 by by ADP american_scientific_journal-4764 37 22 ( ( PUNCT american_scientific_journal-4764 37 23 n n CCONJ american_scientific_journal-4764 37 24 , , PUNCT american_scientific_journal-4764 37 25 p p X american_scientific_journal-4764 37 26 , , PUNCT american_scientific_journal-4764 37 27 q)(e,1 q)(e,1 NOUN american_scientific_journal-4764 37 28 ) ) PUNCT american_scientific_journal-4764 37 29 means mean VERB american_scientific_journal-4764 37 30 . . PUNCT american_scientific_journal-4764 38 1 acknowledgements acknowledgement NOUN american_scientific_journal-4764 38 2 history history NOUN american_scientific_journal-4764 38 3 of of ADP american_scientific_journal-4764 38 4 all all DET american_scientific_journal-4764 38 5 great great ADJ american_scientific_journal-4764 38 6 works work NOUN american_scientific_journal-4764 38 7 into into ADP american_scientific_journal-4764 38 8 witness witness NOUN american_scientific_journal-4764 38 9 that that SCONJ american_scientific_journal-4764 38 10 no no DET american_scientific_journal-4764 38 11 great great ADJ american_scientific_journal-4764 38 12 work work NOUN american_scientific_journal-4764 38 13 was be AUX american_scientific_journal-4764 38 14 ever ever ADV american_scientific_journal-4764 38 15 done do VERB american_scientific_journal-4764 38 16 without without ADP american_scientific_journal-4764 38 17 either either CCONJ american_scientific_journal-4764 38 18 active active ADJ american_scientific_journal-4764 38 19 or or CCONJ american_scientific_journal-4764 38 20 passive passive ADJ american_scientific_journal-4764 38 21 support support NOUN american_scientific_journal-4764 38 22 of of ADP american_scientific_journal-4764 38 23 a a DET american_scientific_journal-4764 38 24 person person NOUN american_scientific_journal-4764 38 25 β€˜ ' PUNCT american_scientific_journal-4764 38 26 surrounding surround VERB american_scientific_journal-4764 38 27 and and CCONJ american_scientific_journal-4764 38 28 one one NUM american_scientific_journal-4764 38 29 ’s ’s PART american_scientific_journal-4764 38 30 close close ADJ american_scientific_journal-4764 38 31 quarters quarter NOUN american_scientific_journal-4764 38 32 . . PUNCT american_scientific_journal-4764 39 1 thus thus ADV american_scientific_journal-4764 39 2 is be AUX american_scientific_journal-4764 39 3 it it PRON american_scientific_journal-4764 39 4 not not PART american_scientific_journal-4764 39 5 hard hard ADJ american_scientific_journal-4764 39 6 to to PART american_scientific_journal-4764 39 7 conclude conclude VERB american_scientific_journal-4764 39 8 how how SCONJ american_scientific_journal-4764 39 9 active active ADJ american_scientific_journal-4764 39 10 assistance assistance NOUN american_scientific_journal-4764 39 11 from from ADP american_scientific_journal-4764 39 12 senior senior ADJ american_scientific_journal-4764 39 13 could could AUX american_scientific_journal-4764 39 14 positively positively ADV american_scientific_journal-4764 39 15 impact impact VERB american_scientific_journal-4764 39 16 the the DET american_scientific_journal-4764 39 17 execution execution NOUN american_scientific_journal-4764 39 18 of of ADP american_scientific_journal-4764 39 19 the the DET american_scientific_journal-4764 39 20 project project NOUN american_scientific_journal-4764 39 21 . . PUNCT american_scientific_journal-4764 40 1 i i PRON american_scientific_journal-4764 40 2 am be AUX american_scientific_journal-4764 40 3 highly highly ADV american_scientific_journal-4764 40 4 thankful thankful ADJ american_scientific_journal-4764 40 5 to to ADP american_scientific_journal-4764 40 6 our our PRON american_scientific_journal-4764 40 7 faculty faculty NOUN american_scientific_journal-4764 40 8 head head NOUN american_scientific_journal-4764 40 9 a.j a.j PROPN american_scientific_journal-4764 40 10 . . PROPN american_scientific_journal-4764 40 11 khan khan PROPN american_scientific_journal-4764 40 12 sir sir PROPN american_scientific_journal-4764 40 13 for for ADP american_scientific_journal-4764 40 14 his his PRON american_scientific_journal-4764 40 15 active active ADJ american_scientific_journal-4764 40 16 guidance guidance NOUN american_scientific_journal-4764 40 17 throughout throughout ADP american_scientific_journal-4764 40 18 the the DET american_scientific_journal-4764 40 19 completion completion NOUN american_scientific_journal-4764 40 20 of of ADP american_scientific_journal-4764 40 21 the the DET american_scientific_journal-4764 40 22 paper paper NOUN american_scientific_journal-4764 40 23 . . PUNCT american_scientific_journal-4764 41 1 last last ADJ american_scientific_journal-4764 41 2 but but CCONJ american_scientific_journal-4764 41 3 not not PART american_scientific_journal-4764 41 4 the the DET american_scientific_journal-4764 41 5 least least ADJ american_scientific_journal-4764 41 6 , , PUNCT american_scientific_journal-4764 41 7 i i PRON american_scientific_journal-4764 41 8 would would AUX american_scientific_journal-4764 41 9 also also ADV american_scientific_journal-4764 41 10 want want VERB american_scientific_journal-4764 41 11 to to PART american_scientific_journal-4764 41 12 extend extend VERB american_scientific_journal-4764 41 13 my my PRON american_scientific_journal-4764 41 14 appreciation appreciation NOUN american_scientific_journal-4764 41 15 to to ADP american_scientific_journal-4764 41 16 those those PRON american_scientific_journal-4764 41 17 who who PRON american_scientific_journal-4764 41 18 could could AUX american_scientific_journal-4764 41 19 not not PART american_scientific_journal-4764 41 20 be be AUX american_scientific_journal-4764 41 21 mentioned mention VERB american_scientific_journal-4764 41 22 here here ADV american_scientific_journal-4764 41 23 but but CCONJ american_scientific_journal-4764 41 24 have have AUX american_scientific_journal-4764 41 25 well well ADV american_scientific_journal-4764 41 26 played play VERB american_scientific_journal-4764 41 27 their their PRON american_scientific_journal-4764 41 28 role role NOUN american_scientific_journal-4764 41 29 in in ADP american_scientific_journal-4764 41 30 inspire inspire VERB american_scientific_journal-4764 41 31 me i PRON american_scientific_journal-4764 41 32 behind behind ADP american_scientific_journal-4764 41 33 the the DET american_scientific_journal-4764 41 34 certain certain ADJ american_scientific_journal-4764 41 35 . . PUNCT american_scientific_journal-4764 42 1 references reference NOUN american_scientific_journal-4764 42 2 [ [ X american_scientific_journal-4764 42 3 1 1 NUM american_scientific_journal-4764 42 4 ] ] X american_scientific_journal-4764 42 5 g.h g.h X american_scientific_journal-4764 42 6 . . NOUN american_scientific_journal-4764 42 7 hardy hardy ADJ american_scientific_journal-4764 42 8 , , PUNCT american_scientific_journal-4764 42 9 divergent divergent ADJ american_scientific_journal-4764 42 10 series series NOUN american_scientific_journal-4764 42 11 , , PUNCT american_scientific_journal-4764 42 12 oxford oxford PROPN american_scientific_journal-4764 42 13 university university PROPN american_scientific_journal-4764 42 14 press press NOUN american_scientific_journal-4764 42 15 , , PUNCT american_scientific_journal-4764 42 16 oxford oxford PROPN american_scientific_journal-4764 42 17 , , PUNCT american_scientific_journal-4764 42 18 uk,1st uk,1st ADJ american_scientific_journal-4764 42 19 edition edition NOUN american_scientific_journal-4764 42 20 , , PUNCT american_scientific_journal-4764 42 21 1949 1949 NUM american_scientific_journal-4764 42 22 . . PUNCT american_scientific_journal-4764 43 1 [ [ X american_scientific_journal-4764 43 2 2 2 NUM american_scientific_journal-4764 43 3 ] ] PUNCT american_scientific_journal-4764 43 4 d. d. NOUN american_scientific_journal-4764 43 5 borwein borwein PROPN american_scientific_journal-4764 43 6 , , PUNCT american_scientific_journal-4764 43 7 on on ADP american_scientific_journal-4764 43 8 product product NOUN american_scientific_journal-4764 43 9 of of ADP american_scientific_journal-4764 43 10 sequences sequence NOUN american_scientific_journal-4764 43 11 , , PUNCT american_scientific_journal-4764 43 12 journal journal NOUN american_scientific_journal-4764 43 13 of of ADP american_scientific_journal-4764 43 14 the the DET american_scientific_journal-4764 43 15 london london PROPN american_scientific_journal-4764 43 16 mathematical mathematical ADJ american_scientific_journal-4764 43 17 society society NOUN american_scientific_journal-4764 43 18 , , PUNCT american_scientific_journal-4764 43 19 vol.33pp.352 vol.33pp.352 NOUN american_scientific_journal-4764 43 20 - - PUNCT american_scientific_journal-4764 43 21 357 357 NUM american_scientific_journal-4764 43 22 , , PUNCT american_scientific_journal-4764 43 23 ( ( PUNCT american_scientific_journal-4764 43 24 1958 1958 NUM american_scientific_journal-4764 43 25 ) ) PUNCT american_scientific_journal-4764 43 26 . . PUNCT american_scientific_journal-4764 44 1 [ [ X american_scientific_journal-4764 44 2 3 3 NUM american_scientific_journal-4764 44 3 ] ] PUNCT american_scientific_journal-4764 44 4 a. a. NOUN american_scientific_journal-4764 44 5 zygmund zygmund PROPN american_scientific_journal-4764 44 6 , , PUNCT american_scientific_journal-4764 44 7 trigonometric trigonometric ADJ american_scientific_journal-4764 44 8 series series NOUN american_scientific_journal-4764 44 9 . . PUNCT american_scientific_journal-4764 45 1 vol vol NOUN american_scientific_journal-4764 45 2 . . PUNCT american_scientific_journal-4764 46 1 i i PRON american_scientific_journal-4764 46 2 cambridge cambridge PROPN american_scientific_journal-4764 46 3 university university PROPN american_scientific_journal-4764 46 4 press press PROPN american_scientific_journal-4764 46 5 , , PUNCT american_scientific_journal-4764 46 6 cambridge cambridge PROPN american_scientific_journal-4764 46 7 , , PUNCT american_scientific_journal-4764 46 8 uk uk PROPN american_scientific_journal-4764 46 9 , , PUNCT american_scientific_journal-4764 46 10 2nd 2nd ADJ american_scientific_journal-4764 46 11 edition edition NOUN american_scientific_journal-4764 46 12 , , PUNCT american_scientific_journal-4764 46 13 1959 1959 NUM american_scientific_journal-4764 46 14 . . PUNCT american_scientific_journal-4764 47 1 [ [ X american_scientific_journal-4764 47 2 4 4 X american_scientific_journal-4764 47 3 ] ] X american_scientific_journal-4764 47 4 p. p. PROPN american_scientific_journal-4764 47 5 chandra chandra PROPN american_scientific_journal-4764 47 6 . . PUNCT american_scientific_journal-4764 48 1 on on ADP american_scientific_journal-4764 48 2 the the DET american_scientific_journal-4764 48 3 degree degree NOUN american_scientific_journal-4764 48 4 of of ADP american_scientific_journal-4764 48 5 approximation approximation NOUN american_scientific_journal-4764 48 6 of of ADP american_scientific_journal-4764 48 7 function function NOUN american_scientific_journal-4764 48 8 belonging belong VERB american_scientific_journal-4764 48 9 to to ADP american_scientific_journal-4764 48 10 lipschitz lipschitz NOUN american_scientific_journal-4764 48 11 class class NOUN american_scientific_journal-4764 48 12 . . PUNCT american_scientific_journal-4764 48 13 labdev labdev PROPN american_scientific_journal-4764 48 14 . . PUNCT american_scientific_journal-4764 49 1 jour jour X american_scientific_journal-4764 49 2 . . PUNCT american_scientific_journal-4764 50 1 science science NOUN american_scientific_journal-4764 50 2 and and CCONJ american_scientific_journal-4764 50 3 technology technology NOUN american_scientific_journal-4764 50 4 , , PUNCT american_scientific_journal-4764 50 5 13a 13a PROPN american_scientific_journal-4764 50 6 ( ( PUNCT american_scientific_journal-4764 50 7 1975 1975 NUM american_scientific_journal-4764 50 8 ) ) PUNCT american_scientific_journal-4764 50 9 , , PUNCT american_scientific_journal-4764 50 10 181 181 NUM american_scientific_journal-4764 50 11 - - SYM american_scientific_journal-4764 50 12 183 183 NUM american_scientific_journal-4764 50 13 . . PUNCT american_scientific_journal-4764 51 1 [ [ X american_scientific_journal-4764 51 2 5 5 NUM american_scientific_journal-4764 51 3 ] ] PUNCT american_scientific_journal-4764 51 4 p. p. PROPN american_scientific_journal-4764 51 5 chandra chandra PROPN american_scientific_journal-4764 51 6 . . PUNCT american_scientific_journal-4764 52 1 on on ADP american_scientific_journal-4764 52 2 the the DET american_scientific_journal-4764 52 3 degree degree NOUN american_scientific_journal-4764 52 4 of of ADP american_scientific_journal-4764 52 5 approximation approximation NOUN american_scientific_journal-4764 52 6 of of ADP american_scientific_journal-4764 52 7 continuous continuous ADJ american_scientific_journal-4764 52 8 functions function NOUN american_scientific_journal-4764 52 9 commun commun X american_scientific_journal-4764 52 10 . . PUNCT american_scientific_journal-4764 53 1 fac fac PROPN american_scientific_journal-4764 53 2 . . PUNCT american_scientific_journal-4764 53 3 sci sci PROPN american_scientific_journal-4764 53 4 . . PROPN american_scientific_journal-4764 53 5 univ univ PROPN american_scientific_journal-4764 53 6 . . PUNCT american_scientific_journal-4764 54 1 ankara ankara PROPN american_scientific_journal-4764 54 2 ser ser PROPN american_scientific_journal-4764 54 3 a a PRON american_scientific_journal-4764 54 4 , , PUNCT american_scientific_journal-4764 54 5 30(1981),7 30(1981),7 NOUN american_scientific_journal-4764 54 6 - - SYM american_scientific_journal-4764 54 7 16 16 NUM american_scientific_journal-4764 54 8 . . PUNCT american_scientific_journal-4764 55 1 [ [ X american_scientific_journal-4764 55 2 6 6 NUM american_scientific_journal-4764 55 3 ] ] X american_scientific_journal-4764 55 4 k.ikeno k.ikeno PROPN american_scientific_journal-4764 55 5 , , PUNCT american_scientific_journal-4764 55 6 lebesgue lebesgue NOUN american_scientific_journal-4764 55 7 constant constant ADJ american_scientific_journal-4764 55 8 for for ADP american_scientific_journal-4764 55 9 a a DET american_scientific_journal-4764 55 10 family family NOUN american_scientific_journal-4764 55 11 of of ADP american_scientific_journal-4764 55 12 summability summability NOUN american_scientific_journal-4764 55 13 method method NOUN american_scientific_journal-4764 55 14 , , PUNCT american_scientific_journal-4764 55 15 tohoku tohoku PROPN american_scientific_journal-4764 55 16 math math PROPN american_scientific_journal-4764 55 17 . . PUNCT american_scientific_journal-4764 56 1 j. j. PROPN american_scientific_journal-4764 56 2 ,17(1965),250 ,17(1965),250 PROPN american_scientific_journal-4764 56 3 - - PUNCT american_scientific_journal-4764 56 4 265 265 NUM american_scientific_journal-4764 56 5 . . PUNCT american_scientific_journal-4764 57 1 [ [ X american_scientific_journal-4764 57 2 7 7 NUM american_scientific_journal-4764 57 3 ] ] SYM american_scientific_journal-4764 57 4 b.kuttner b.kuttner NOUN american_scientific_journal-4764 57 5 , , PUNCT american_scientific_journal-4764 57 6 c.t.rjajagopal c.t.rjajagopal ADJ american_scientific_journal-4764 57 7 , , PUNCT american_scientific_journal-4764 57 8 and and CCONJ american_scientific_journal-4764 57 9 m.s.rangachari m.s.rangachari PROPN american_scientific_journal-4764 57 10 , , PUNCT american_scientific_journal-4764 57 11 tauberian tauberian ADJ american_scientific_journal-4764 57 12 convergence convergence NOUN american_scientific_journal-4764 57 13 theorems theorem VERB american_scientific_journal-4764 57 14 for for ADP american_scientific_journal-4764 57 15 summability summability NOUN american_scientific_journal-4764 57 16 methods method NOUN american_scientific_journal-4764 57 17 of of ADP american_scientific_journal-4764 57 18 the the DET american_scientific_journal-4764 57 19 karamata karamata ADJ american_scientific_journal-4764 57 20 family family NOUN american_scientific_journal-4764 57 21 , , PUNCT american_scientific_journal-4764 57 22 j.indian j.indian ADJ american_scientific_journal-4764 57 23 math math NOUN american_scientific_journal-4764 57 24 . . PUNCT american_scientific_journal-4764 58 1 soc soc NOUN american_scientific_journal-4764 58 2 . . PUNCT american_scientific_journal-4764 59 1 44(1980),23 44(1980),23 PROPN american_scientific_journal-4764 59 2 - - PUNCT american_scientific_journal-4764 59 3 38 38 NUM american_scientific_journal-4764 59 4 . . PUNCT american_scientific_journal-4764 60 1 [ [ X american_scientific_journal-4764 60 2 8 8 NUM american_scientific_journal-4764 60 3 ] ] PUNCT american_scientific_journal-4764 60 4 a.meir a.meir ADJ american_scientific_journal-4764 60 5 , , PUNCT american_scientific_journal-4764 60 6 tauberian tauberian ADJ american_scientific_journal-4764 60 7 constants constant NOUN american_scientific_journal-4764 60 8 for for ADP american_scientific_journal-4764 60 9 a a DET american_scientific_journal-4764 60 10 family family NOUN american_scientific_journal-4764 60 11 of of ADP american_scientific_journal-4764 60 12 transformations transformation NOUN american_scientific_journal-4764 60 13 , , PUNCT american_scientific_journal-4764 60 14 annals annal NOUN american_scientific_journal-4764 60 15 of of ADP american_scientific_journal-4764 60 16 math math NOUN american_scientific_journal-4764 60 17 . . PUNCT american_scientific_journal-4764 61 1 78(1963 78(1963 NUM american_scientific_journal-4764 61 2 ) ) PUNCT american_scientific_journal-4764 61 3 , , PUNCT american_scientific_journal-4764 61 4 594 594 NUM american_scientific_journal-4764 61 5 - - SYM american_scientific_journal-4764 61 6 599 599 NUM american_scientific_journal-4764 61 7 [ [ X american_scientific_journal-4764 61 8 9 9 NUM american_scientific_journal-4764 61 9 ] ] SYM american_scientific_journal-4764 61 10 u.k.shrivastava u.k.shrivastava NOUN american_scientific_journal-4764 61 11 and and CCONJ american_scientific_journal-4764 61 12 s.l.varma s.l.varma NOUN american_scientific_journal-4764 61 13 , , PUNCT american_scientific_journal-4764 61 14 on on ADP american_scientific_journal-4764 61 15 the the DET american_scientific_journal-4764 61 16 degree degree NOUN american_scientific_journal-4764 61 17 of of ADP american_scientific_journal-4764 61 18 approximation approximation NOUN american_scientific_journal-4764 61 19 of of ADP american_scientific_journal-4764 61 20 function function NOUN american_scientific_journal-4764 61 21 belonging belong VERB american_scientific_journal-4764 61 22 to to AUX american_scientific_journal-4764 61 23 lipschitz lipschitz VERB american_scientific_journal-4764 61 24 class class NOUN american_scientific_journal-4764 61 25 by by ADP american_scientific_journal-4764 61 26 ( ( PUNCT american_scientific_journal-4764 61 27 e e NOUN american_scientific_journal-4764 61 28 , , PUNCT american_scientific_journal-4764 61 29 c c NOUN american_scientific_journal-4764 61 30 ) ) PUNCT american_scientific_journal-4764 61 31 means mean NOUN american_scientific_journal-4764 61 32 , , PUNCT american_scientific_journal-4764 61 33 tamkang tamkang PROPN american_scientific_journal-4764 61 34 journal journal PROPN american_scientific_journal-4764 61 35 of of ADP american_scientific_journal-4764 61 36 maths math NOUN american_scientific_journal-4764 61 37 , , PUNCT american_scientific_journal-4764 61 38 26,no.3(1965),97 26,no.3(1965),97 PROPN american_scientific_journal-4764 61 39 - - SYM american_scientific_journal-4764 61 40 101 101 NUM american_scientific_journal-4764 61 41 . . PUNCT