id sid eid entity type easat-2091 1 1 2576-8484 DATE easat-2091 1 2 8 CARDINAL easat-2091 1 3 6 CARDINAL easat-2091 1 4 394 CARDINAL easat-2091 1 5 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 1 6 2024 CARDINAL easat-2091 1 7 2024 CARDINAL easat-2091 1 8 o.osman@qu.edu.sa CARDINAL easat-2091 1 9 osman abdalla adam osman1 PERSON easat-2091 1 10 1department CARDINAL easat-2091 1 11 51452 DATE easat-2091 1 12 saudi arabia GPE easat-2091 1 13 o.osman@qu.edu.sa CARDINAL easat-2091 5 1 bergman PERSON easat-2091 6 1 1 CARDINAL easat-2091 6 2 2 CARDINAL easat-2091 6 3 analytically𝐷. GPE easat-2091 6 4 bergmann PERSON easat-2091 6 5 2 CARDINAL easat-2091 6 6 subspace𝐿2(𝐷 ORG easat-2091 6 7 𝐿2(𝐷 GPE easat-2091 6 8 2 CARDINAL easat-2091 6 9 1,6 CARDINAL easat-2091 8 1 + 1 DATE easat-2091 8 2 + 1 DATE easat-2091 8 3 beforeπ‘‡πœ‘+1𝑓 = 𝑃{(πœ‘ ORG easat-2091 11 1 zero CARDINAL easat-2091 12 1 1 CARDINAL easat-2091 12 2 𝐷. NORP easat-2091 12 3 𝐷. NORP easat-2091 12 4 π‘Ž(πœ‘ GPE easat-2091 12 5 + 1 DATE easat-2091 13 1 + 1 DATE easat-2091 14 1 1 CARDINAL easat-2091 14 2 2,7 CARDINAL easat-2091 15 1 brown PERSON easat-2091 16 1 2 CARDINAL easat-2091 18 1 1 CARDINAL easat-2091 18 2 3.8 CARDINAL easat-2091 19 1 1 CARDINAL easat-2091 20 1 1 CARDINAL easat-2091 21 1 395 CARDINAL easat-2091 21 2 2576-8484 DATE easat-2091 21 3 8 CARDINAL easat-2091 22 1 6 CARDINAL easat-2091 22 2 394 CARDINAL easat-2091 22 3 2024 CARDINAL easat-2091 22 4 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 22 5 2024 CARDINAL easat-2091 22 6 1 CARDINAL easat-2091 23 1 paul bourdon PERSON easat-2091 23 2 two CARDINAL easat-2091 23 3 in𝐿∞(𝐷 ORG easat-2091 24 1 𝐿∞(𝐷 GPE easat-2091 24 2 2 CARDINAL easat-2091 27 1 one CARDINAL easat-2091 27 2 one CARDINAL easat-2091 29 1 ∈ ORG easat-2091 30 1 1 CARDINAL easat-2091 30 2 ∈ ORG easat-2091 30 3 ∈ ORG easat-2091 30 4 𝐢(𝐷 PERSON easat-2091 30 5 2πœ‹ CARDINAL easat-2091 30 6 1 CARDINAL easat-2091 30 7 ∈ ORG easat-2091 31 1 0 CARDINAL easat-2091 31 2 1 CARDINAL easat-2091 33 1 ∈ ORG easat-2091 34 1 4,6 CARDINAL easat-2091 34 2 ∈ ORG easat-2091 34 3 𝐢(𝐷 PERSON easat-2091 35 1 2 CARDINAL easat-2091 35 2 ∈ ORG easat-2091 36 1 2πœ‹ CARDINAL easat-2091 37 1 2 CARDINAL easat-2091 37 2 ∈ ORG easat-2091 38 1 in𝐢(𝐷 NORP easat-2091 38 2 ∈ ORG easat-2091 39 1 ∈ ORG easat-2091 39 2 0 CARDINAL easat-2091 39 3 1 CARDINAL easat-2091 40 1 ∈ ORG easat-2091 43 1 ∈ ORG easat-2091 44 1 𝐢(𝐷 PERSON easat-2091 44 2 4,5 CARDINAL easat-2091 47 1 ∈ 𝐢(𝐷 PERSON easat-2091 47 2 denoted𝑅(𝑒 ORG easat-2091 47 3 2πœ‹ CARDINAL easat-2091 47 4 2πœ‹ CARDINAL easat-2091 47 5 3 CARDINAL easat-2091 47 6 1 CARDINAL easat-2091 47 7 ∈ ORG easat-2091 47 8 2 CARDINAL easat-2091 47 9 ∈ 𝐢(𝐷 PERSON easat-2091 50 1 396 CARDINAL easat-2091 50 2 2576-8484 DATE easat-2091 51 1 8 CARDINAL easat-2091 52 1 6 CARDINAL easat-2091 52 2 394 CARDINAL easat-2091 52 3 2024 CARDINAL easat-2091 52 4 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 52 5 2024 CARDINAL easat-2091 52 6 4 CARDINAL easat-2091 52 7 ∈ ORG easat-2091 52 8 ∈ ORG easat-2091 52 9 5 CARDINAL easat-2091 52 10 ∈ ORG easat-2091 54 1 5 CARDINAL easat-2091 56 1 letβ„Ž ∈ ORG easat-2091 57 1 ∈ ORG easat-2091 58 1 ∈ ORG easat-2091 62 1 2πœ‹ CARDINAL easat-2091 62 2 2πœ‹ CARDINAL easat-2091 62 3 πœ‹ PERSON easat-2091 62 4 6 CARDINAL easat-2091 63 1 ∈ ORG easat-2091 63 2 π‘“πœƒ(𝑀 ORG easat-2091 63 3 ∈ ORG easat-2091 63 4 1βˆ’ CARDINAL easat-2091 63 5 |(π‘“πœƒ ORG easat-2091 64 1 1βˆ’|𝛽|2 CARDINAL easat-2091 64 2 |(π‘“πœƒ ORG easat-2091 66 1 0,2πœ‹ CARDINAL easat-2091 66 2 β„Ž(g(𝑀)𝑒 π‘–πœƒ))| 𝑑𝐴(𝑀 ORG easat-2091 67 1 2πœ‹ CARDINAL easat-2091 69 1 |(π‘“πœƒ ORG easat-2091 71 1 2πœ‹ CARDINAL easat-2091 71 2 0 ≀ MONEY easat-2091 72 1 Β° LANGUAGE easat-2091 75 1 2πœ‹ CARDINAL easat-2091 78 1 2πœ‹ CARDINAL easat-2091 80 1 2πœ‹ CARDINAL easat-2091 81 1 2πœ‹ CARDINAL easat-2091 82 1 Β° LANGUAGE easat-2091 84 1 4,5 CARDINAL easat-2091 85 1 𝑣(0 PERSON easat-2091 86 1 Β° TIME easat-2091 86 2 2πœ‹ CARDINAL easat-2091 87 1 0 CARDINAL easat-2091 87 2 1 CARDINAL easat-2091 87 3 ∈ ORG easat-2091 89 1 4 CARDINAL easat-2091 89 2 𝐷. NORP easat-2091 89 3 2 CARDINAL easat-2091 91 1 patrick ahern PERSON easat-2091 91 2 walter rudin PERSON easat-2091 91 3 2 CARDINAL easat-2091 91 4 3 CARDINAL easat-2091 92 1 3 CARDINAL easat-2091 92 2 𝑉 ∈ ORG easat-2091 93 1 ∩ 𝐿/[0,1 ORG easat-2091 93 2 397 CARDINAL easat-2091 93 3 2576-8484 DATE easat-2091 94 1 8 CARDINAL easat-2091 95 1 6 CARDINAL easat-2091 95 2 394 CARDINAL easat-2091 95 3 2024 CARDINAL easat-2091 95 4 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 95 5 2024 CARDINAL easat-2091 96 1 𝑣(𝑑 PERSON easat-2091 96 2 1 CARDINAL easat-2091 96 3 1+𝑑𝑠 CARDINAL easat-2091 96 4 1βˆ’π‘‘π‘ )2 CARDINAL easat-2091 96 5 0,1 CARDINAL easat-2091 96 6 7 CARDINAL easat-2091 96 7 𝐢(𝐷 PERSON easat-2091 97 1 first ORDINAL easat-2091 97 2 𝐢(𝐷 PERSON easat-2091 99 1 0,1 CARDINAL easat-2091 99 2 ∈ ORG easat-2091 99 3 β„Žπ›Ό(𝑧 CARDINAL easat-2091 99 4 1βˆ’π›Όπ‘§ CARDINAL easat-2091 101 1 βˆ«π·π‘£(𝑀 DATE easat-2091 103 1 1 CARDINAL easat-2091 103 2 1 CARDINAL easat-2091 104 1 4 2πœ‹ CARDINAL easat-2091 104 2 2πœ‹ CARDINAL easat-2091 104 3 1 CARDINAL easat-2091 104 4 1βˆ’π‘Ž2π‘Ÿ2)3 1 DATE easat-2091 104 5 1 CARDINAL easat-2091 104 6 1+π‘Ž2π‘Ÿ2 CARDINAL easat-2091 104 7 1βˆ’π‘Ž2𝑠)3 1 DATE easat-2091 104 8 0 CARDINAL easat-2091 104 9 1 CARDINAL easat-2091 104 10 7 CARDINAL easat-2091 105 1 0,1 CARDINAL easat-2091 107 1 ∈ ORG easat-2091 108 1 Β° TIME easat-2091 108 2 2 CARDINAL easat-2091 109 1 Β° TIME easat-2091 109 2 Β° TIME easat-2091 109 3 ∈ ORG easat-2091 109 4 lemma 2 ORG easat-2091 110 1 3 CARDINAL easat-2091 110 2 ∈ ORG easat-2091 110 3 π‘ˆβ„Ž ORG easat-2091 110 4 π‘ˆβ„Ž ORG easat-2091 110 5 2 CARDINAL easat-2091 110 6 2 CARDINAL easat-2091 111 1 1 CARDINAL easat-2091 111 2 4 CARDINAL easat-2091 111 3 ∈ ORG easat-2091 111 4 let(πœ‘ + 1 FAC easat-2091 113 1 𝐿2(𝐷 GPE easat-2091 113 2 byπ‘‰β„Žπ‘“ GPE easat-2091 114 1 𝐿2(𝐷 GPE easat-2091 116 1 2 CARDINAL easat-2091 116 2 2 CARDINAL easat-2091 116 3 2 CARDINAL easat-2091 116 4 + 1)Β°β„Ž(𝑓° DATE easat-2091 116 5 + 1) DATE easat-2091 116 6 Β° TIME easat-2091 118 1 + 1𝑓 DATE easat-2091 118 2 + 1and DATE easat-2091 118 3 π‘ˆβ„Ž ORG easat-2091 120 1 that𝐻1(𝐷 CARDINAL easat-2091 121 1 2 CARDINAL easat-2091 122 1 1 CARDINAL easat-2091 122 2 ∈ ORG easat-2091 123 1 2 CARDINAL easat-2091 123 2 ∈ ORG easat-2091 124 1 1 CARDINAL easat-2091 124 2 1 CARDINAL easat-2091 126 1 π‘‡πœ‘ + 1 and π‘‡πœ“ + 1 PRODUCT easat-2091 126 2 2 CARDINAL easat-2091 126 3 398 CARDINAL easat-2091 126 4 2576-8484 DATE easat-2091 126 5 8 CARDINAL easat-2091 126 6 6 CARDINAL easat-2091 126 7 394 CARDINAL easat-2091 126 8 2024 CARDINAL easat-2091 126 9 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 126 10 2024 CARDINAL easat-2091 128 1 π‘‡πœ‘ + 1Μ…Μ… ORG easat-2091 131 1 π‘‡πœ‘ + 1Μ…Μ… ORG easat-2091 133 1 + 1 DATE easat-2091 134 1 𝑏 ∈ 𝐢 PERSON easat-2091 134 2 π‘Ž(πœ‘ GPE easat-2091 134 3 + 1 DATE easat-2091 135 1 + 1 DATE easat-2091 135 2 + 1 DATE easat-2091 136 1 πΏπ‘Ž 2 LAW easat-2091 137 1 π‘‡πœ‘ + 1π‘‡πœ“ ORG easat-2091 139 1 𝑏 β‰  0 PERSON easat-2091 140 1 π‘‡πœ‘ + 1 and π‘‡πœ“ + 1 PRODUCT easat-2091 141 1 1 CARDINAL easat-2091 143 1 πœ‘ + 1 LOC easat-2091 143 2 𝑓1 ORG easat-2091 144 1 𝑓1 + 𝑓2Μ… ORG easat-2091 144 2 9 CARDINAL easat-2091 144 3 𝑓1 ORG easat-2091 144 4 𝐻2(𝐷 GPE easat-2091 145 1 𝐷. NORP easat-2091 156 1 βŒ©π‘‡πœ‘ + 1π‘‡πœ“ PERSON easat-2091 157 1 1βŒͺ QUANTITY easat-2091 159 1 πœ‹[𝑓2(0)g1(0 DATE easat-2091 161 1 10 CARDINAL easat-2091 161 2 βŒ©π‘‡πœ‘ + 1π‘‡πœ“ PERSON easat-2091 161 3 π‘‡πœ‘ + 1π‘‡πœ“ ORG easat-2091 162 1 10 CARDINAL easat-2091 162 2 βŒ©π‘‡πœ‘ + 1π‘‡πœ“ PERSON easat-2091 163 1 11 CARDINAL easat-2091 163 2 ∈ ORG easat-2091 166 1 π‘ˆβ„Ž βˆ— ORG easat-2091 166 2 π‘ˆβ„Ž ORG easat-2091 166 3 1 CARDINAL easat-2091 168 1 4 CARDINAL easat-2091 170 1 + 1)βˆ—β„Ž DATE easat-2091 170 2 12 CARDINAL easat-2091 170 3 13 CARDINAL easat-2091 170 4 eq(11 DATE easat-2091 171 1 π‘‡πœ‘ + 1π‘‡πœ“ ORG easat-2091 172 1 eq(11 CARDINAL easat-2091 172 2 πœ‹ = 𝑓2Μ…(β„Ž(0))g1(β„Ž(0 PERSON easat-2091 173 1 𝑓1(β„Ž(0))g2Μ…Μ… CARDINAL easat-2091 175 1 399 CARDINAL easat-2091 175 2 2576-8484 DATE easat-2091 175 3 8 CARDINAL easat-2091 176 1 6 CARDINAL easat-2091 176 2 394 CARDINAL easat-2091 176 3 2024 CARDINAL easat-2091 176 4 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 176 5 2024 CARDINAL easat-2091 176 6 lemma 2 ORG easat-2091 176 7 ∈ ORG easat-2091 177 1 taylor PERSON easat-2091 177 2 eq(13 ORG easat-2091 177 3 𝐻2(𝐷 GPE easat-2091 179 1 2 ∞ QUANTITY easat-2091 179 2 2 ∞ QUANTITY easat-2091 179 3 14 CARDINAL easat-2091 179 4 2πœ‹ CARDINAL easat-2091 180 1 2πœ‹ CARDINAL easat-2091 180 2 𝛽𝑛|𝑍| ORG easat-2091 181 1 2𝑛 CARDINAL easat-2091 182 1 𝐢(𝐷 PERSON easat-2091 182 2 ∈ ORG easat-2091 182 3 ∈ ORG easat-2091 183 1 ∈ ORG easat-2091 185 1 πœ•π‘§ CARDINAL easat-2091 185 2 πœ•π‘§ CARDINAL easat-2091 185 3 πœ•π‘§ CARDINAL easat-2091 185 4 1 2 CARDINAL easat-2091 185 5 πœ•π‘₯ CARDINAL easat-2091 185 6 πœ•π‘§ CARDINAL easat-2091 185 7 1 2 CARDINAL easat-2091 185 8 πœ•π‘₯ + DATE easat-2091 186 1 πœ•π‘§ CARDINAL easat-2091 186 2 πœ•π‘§ CARDINAL easat-2091 186 3 0 CARDINAL easat-2091 186 4 πœ•π‘§ CARDINAL easat-2091 186 5 0 CARDINAL easat-2091 186 6 πœ•π‘§ CARDINAL easat-2091 187 1 πœ•π‘§ CARDINAL easat-2091 187 2 πœ•π‘§ CARDINAL easat-2091 188 1 πœ•π‘₯2 + πœ•2 πœ•π‘¦2 GPE easat-2091 188 2 4 CARDINAL easat-2091 188 3 πœ•π‘§ CARDINAL easat-2091 188 4 πœ•π‘§ CARDINAL easat-2091 188 5 0 CARDINAL easat-2091 188 6 4 CARDINAL easat-2091 188 7 πœ•π‘§ CARDINAL easat-2091 189 1 4 CARDINAL easat-2091 189 2 πœ•π‘§ CARDINAL easat-2091 189 3 πœ•π‘§ CARDINAL easat-2091 189 4 4 πœ• πœ•π‘§ MONEY easat-2091 189 5 βˆ’ f1 PERSON easat-2091 190 1 15 CARDINAL easat-2091 191 1 g1 ORG easat-2091 191 2 0 CARDINAL easat-2091 192 1 𝑓1 ORG easat-2091 193 1 0 CARDINAL easat-2091 195 1 g1 ORG easat-2091 195 2 0 CARDINAL easat-2091 195 3 15 CARDINAL easat-2091 195 4 g1 ORG easat-2091 196 1 𝑓1 ORG easat-2091 196 2 g1 ORG easat-2091 196 3 ∈ 𝐢. thus𝑓1 PERSON easat-2091 196 4 𝑐g1 ORG easat-2091 196 5 𝑓1 βˆ’ 𝑐g1 PERSON easat-2091 196 6 2 CARDINAL easat-2091 196 7 2 CARDINAL easat-2091 196 8 400 CARDINAL easat-2091 196 9 2576-8484 DATE easat-2091 196 10 8 CARDINAL easat-2091 196 11 6 CARDINAL easat-2091 196 12 394 CARDINAL easat-2091 196 13 2024 CARDINAL easat-2091 196 14 10.55214/25768484.v8i6.2091 CARDINAL easat-2091 196 15 2024 CARDINAL easat-2091 196 16 1 CARDINAL easat-2091 198 1 theorem1 PERSON easat-2091 198 2 π‘‡πœ‘ + 1is ORG easat-2091 199 1 5 CARDINAL easat-2091 200 1 first ORDINAL easat-2091 200 2 ∈ 𝐢 ORG easat-2091 200 3 + 1 DATE easat-2091 202 1 + 1)+𝛽 DATE easat-2091 202 2 π‘‡πœ‘ + 1 PRODUCT easat-2091 203 1 π‘‡πœ‘ + 1 PRODUCT easat-2091 205 1 π‘‡πœ‘ + 1π‘‡πœ‘ + 1Μ…Μ… ORG easat-2091 208 1 1 CARDINAL easat-2091 208 2 ∈ 𝐢 ORG easat-2091 208 3 π‘Ž(πœ‘ GPE easat-2091 209 1 𝐷. NORP easat-2091 210 1 2024 CARDINAL easat-2091 211 1 https://creativecommons.org/licenses/by/4.0/ WORK_OF_ART easat-2091 212 1 1 CARDINAL easat-2091 213 1 14 (1991 DATE easat-2091 214 1 2 CARDINAL easat-2091 214 2 arlen brown PERSON easat-2091 214 3 r. halmas PERSON easat-2091 214 4 j. reine angew PERSON easat-2091 216 1 213(1964 CARDINAL easat-2091 216 2 89 CARDINAL easat-2091 217 1 3 CARDINAL easat-2091 217 2 pamela gorkin PERSON easat-2091 217 3 algebras ORG easat-2091 218 1 amer PERSON easat-2091 220 1 309 CARDINAL easat-2091 220 2 1988 DATE easat-2091 220 3 711 CARDINAL easat-2091 221 1 4 CARDINAL easat-2091 221 2 walter rudin PERSON easat-2091 221 3 new york GPE easat-2091 221 4 1980 DATE easat-2091 222 1 5 CARDINAL easat-2091 222 2 j. arazy PERSON easat-2091 222 3 s. d. fisher PERSON easat-2091 222 4 j. peeter PERSON easat-2091 222 5 bergman spaces PERSON easat-2091 222 6 american NORP easat-2091 223 1 110 CARDINAL easat-2091 223 2 1988 DATE easat-2091 223 3 989-1053 CARDINAL easat-2091 224 1 6 CARDINAL easat-2091 224 2 s. PERSON easat-2091 224 3 duke j. math PERSON easat-2091 225 1 53(1986 CARDINAL easat-2091 225 2 315 CARDINAL easat-2091 226 1 zheng PERSON easat-2091 226 2 d. NORP easat-2091 226 3 j. funct PERSON easat-2091 229 1 98–120 CARDINAL easat-2091 230 1 8 CARDINAL easat-2091 230 2 s. GPE easat-2091 230 3 chang, y.a.s. PERSON easat-2091 230 4 d. NORP easat-2091 230 5 285–309 CARDINAL