Borel's methods of summability: theory and applications
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford u.a.
Clarendon Press
1994
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| Series: | Oxford mathematical monographs
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 242 S. graph. Darst. |
| ISBN: | 0198535856 |
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MARC
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Record in the Search Index
| _version_ | 1804124508802514944 |
|---|---|
| adam_text | Contents
0 Introduction 1
1 Historical overview 3
2 Summability methods in general 9
2.1 Regularity 9
2.2 Generalized Cesaro summability 16
2.3 Sequence to function methods based on power series 23
2.4 Euler summability 25
3 Borel s methods of summability 27
3.1 Basic definitions 27
3.2 Basic properties of Borel s methods 28
3.2.1 Common properties 28
3.2.2 Interrelationships 29
3.3 Extensions 30
3.3.1 Absolute summability 30
3.3.2 Strong summability 31
3.3.3 Normal and regular summability 31
3.4 Relationships with other methods 32
3.4.1 Euler methods 32
3.4.2 Cesaro and Abel methods 32
3.4.3 Other methods 34
3.5 Abelian theorems 34
3.6 Tauberian theorems 35
4 Relations with the family of circle methods 37
4.1 Euler Knopp summability methods 37
4.2 Ta methods 39
x Contents
4.2.1 Definitions 39
4.2.2 Ta on series 40
4.2.3 Ta and T0 41
4.2.4 Ta and £ 42
4.2.5 Translativity 43
4.3 Meyer Konig s Sa methods 43
4.3.1 Definition 43
4.3.2 Translativity 44
4.3.3 Sa on series 45
4.3.4 Sa and Sp 45
4.3.5 Sa and Ep 46
4.3.6 Function theoretic considerations 46
4.4 Relations of Ta and Sa with Ep and B 48
4.5 Relations of Ep, B, and Sa with TQ 49
4.6 Equivalence of Ep, B, Sa,Ta for bounded sequences 54
4.7 Tauberian theorems 54
5 Generalizations of Borel s methods 57
5.1 First attempts 57
5.2 Mittag Leffler s functions 58
5.3 Borel type methods 59
5.3.1 Definitions 59
5.3.2 Preliminaries 62
5.3.3 Lemmas 64
5.4 Relationships with respect to the parameter a 66
5.5 Abelian relationships with respect to the parameter /3 67
5.5.1 Interrelationships with same type 67
5.5.2 Interrelationships between types 69
5.6 Tauberian relationships with respect to the parameter /? 79
5.6.1 Preliminary results 80
5.6.2 Proofs of the theorems 84
5.7 Extended definitions 85
5.7.1 Results involving strong summability 86
5.7.2 Results involving absolute summability 87
6 Abelian theorems 89
6.1 Introduction 89
6.2 Abelian theorems for ordinary Borel type methods 89
6.3 Abelian theorems for strong Borel type methods 93
6.4 Abelian theorems for absolute Borel type methods 94
Contents xi
7 Tauberian theorems — I 97
7.1 The o theorem 98
7.1.1 Preliminary results 98
7.1.2 Results on Cesaro sums 102
7.1.3 Proof of the o theorem 107
7.2 The O theorem 108
7.2.1 Preliminary results 108
7.2.2 Estimates of some sums as integrals 110
7.2.3 Results on summability (e, c) 112
7.2.4 Two preliminary theorems 118
7.2.5 Proof of the O theorem 121
7.3 Kwee s O theorem 122
7.3.1 Preliminary results 123
7.3.2 Proof of Kwee s O theorem 127
7.3.3 Kwee s O theorem is best possible 129
8 Tauberian theorems II 133
8.1 The slowly decreasing theorem 133
8.1.1 Preliminary results 135
8.2 An equivalence theorem 149
8.3 Proof of the slowly decreasing theorem 157
8.4 Gap theorems 157
9 Relationships with other methods 159
9.1 Product methods with the Cesaro method 159
9.1.1 Product methods 159
9.1.2 Preliminary results 160
9.1.3 Proof of the Cesaro product theorem 162
9.2 Abelian relations with the Abel type methods 164
9.2.1 Review of the definitions 164
9.2.2 Preliminary results 165
9.2.3 Theorems from Borel to Abel 167
9.3 Tauberian relations with the Abel type methods 168
9.3.1 Preliminary results 169
9.3.2 Theorems from Abel to Borel 169
9.4 Tauberian relations with the logarithmic method 175
9.4.1 Preliminary results 175
9.4.2 The logarithmic theorem 177
9.5 Relations with the Lambert method 180
9.5.1 Transformation formulae 183
9.5.2 Essential lemmas 186
xii Contents
9.5.3 Proof of the Lambert theorem 188
10 Applications of Borel s methods 191
10.1 An early application 191
10.2 Laplace transforms 193
10.3 Entire functions and the Borel transform 195
10.3.1 The Phragmen Lindelof indicator function 195
10.3.2 The conjugate indicator diagram 196
10.4 Arithmetical functions 198
10.5 Basic theory 204
References 207
Index 239
|
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| author | Shawyer, Bruce Watson, Bruce |
| author_facet | Shawyer, Bruce Watson, Bruce |
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| ctrlnum | (OCoLC)246703137 (DE-599)BVBBV010116301 |
| dewey-full | 515.243 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.243 |
| dewey-search | 515.243 |
| dewey-sort | 3515.243 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010116301 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:46:47Z |
| institution | BVB |
| isbn | 0198535856 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006717042 |
| oclc_num | 246703137 |
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| owner_facet | DE-12 DE-739 DE-91G DE-BY-TUM DE-11 DE-188 |
| physical | XII, 242 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford mathematical monographs |
| spelling | Shawyer, Bruce Verfasser aut Borel's methods of summability theory and applications Bruce Shawyer and Bruce Watson Oxford u.a. Clarendon Press 1994 XII, 242 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Borel-Summierungsverfahren (DE-588)4146328-6 gnd rswk-swf Borel-Summierungsverfahren (DE-588)4146328-6 s DE-604 Watson, Bruce Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006717042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shawyer, Bruce Watson, Bruce Borel's methods of summability theory and applications Borel-Summierungsverfahren (DE-588)4146328-6 gnd |
| subject_GND | (DE-588)4146328-6 |
| title | Borel's methods of summability theory and applications |
| title_auth | Borel's methods of summability theory and applications |
| title_exact_search | Borel's methods of summability theory and applications |
| title_full | Borel's methods of summability theory and applications Bruce Shawyer and Bruce Watson |
| title_fullStr | Borel's methods of summability theory and applications Bruce Shawyer and Bruce Watson |
| title_full_unstemmed | Borel's methods of summability theory and applications Bruce Shawyer and Bruce Watson |
| title_short | Borel's methods of summability |
| title_sort | borel s methods of summability theory and applications |
| title_sub | theory and applications |
| topic | Borel-Summierungsverfahren (DE-588)4146328-6 gnd |
| topic_facet | Borel-Summierungsverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006717042&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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