Tauberian theory: a century of developments
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Series: | Die Grundlehren der mathematischen Wissenschaften
329 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XV, 483 S. graph. Darst. |
| ISBN: | 354021058X 9783642059193 |
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MARC
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| 245 | 1 | 0 | |a Tauberian theory |b a century of developments |c Jacob Korevaar |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XV, 483 S. |b graph. Darst. | ||
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| 490 | 1 | |a Die Grundlehren der mathematischen Wissenschaften |v 329 | |
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| 650 | 4 | |a Summability theory | |
| 650 | 4 | |a Tauberian theorems | |
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Record in the Search Index
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| adam_text |
Contents
I The Hardy Littlewood Theorems 1
1 Introduction 1
2 Examples of Summability Methods.
Abelian Theorems and Tauberian Question 3
3 Simple Applications of Cesaro, Abel and Borel Summability 6
4 Lambert Summability in Number Theory 8
5 Tauber's Theorems for Abel Summability 10
6 Tauberian Theorem for Cesaro Summability 12
7 Hardy Littlewood Tauberians for Abel Summability 14
8 Tauberians Involving Dirichlet Series 17
9 Tauberians for Borel Summability 18
10 Lambert Tauberian and Prime Number Theorem 18
11 Karamata's Method for Power Series 20
12 Wielandt's Variation on the Method 22
13 Transition from Series to Integrals 24
14 Extension of Tauber's Theorems to Laplace Stieltjes Transforms . 28
15 Hardy Littlewood Type Theorems Involving Laplace Transforms . 29
16 Other Tauberian Conditions: Slowly Decreasing Functions 32
17 Asymptotics for Derivatives 34
18 Integral Tauberians for Cesaro Summability 37
19 The Method of the Monotone Minorant 39
20 Boundedness Theorem Involving a General Kernel Transform 43
21 Laplace Stieltjes and Stieltjes Transform 45
22 General Dirichlet Series 48
23 The High Indices Theorem 50
24 Optimality of Tauberian Conditions 54
25 Tauberian Theorems of Nonstandard Type 58
26 Important Properties of the Zeta Function 62
XII Contents
II Wiener's Theory 65
1 Introduction 65
2 Wiener Problem: Pitt's Form 67
3 Testing Equation for Wiener Kernels 70
4 Original Wiener Problem 72
5 Wiener's Theorem With Additions by Pitt 74
6 Direct Applications of the Testing Equations 77
7 Fourier Analysis of Wiener Kernels 79
8 The Principal Wiener Theorems 82
9 Proof of the Division Theorem 85
10 Wiener Families of Kernels 89
11 Distributional Approach to Wiener Theory 90
12 General Tauberian for Lambert Summability 92
13 Wiener's 'Second Tauberian Theorem' 95
14 A Wiener Theorem for Series 98
15 Extensions 101
16 Discussion of the Tauberian Conditions 103
17 Landau Ingham Asymptotics 106
18 Ingham Summability 110
19 Application of Wiener Theory to Harmonic Functions Ill
III Complex Tauberian Theorems 117
1 Introduction 117
2 A Landau Type Tauberian for Dirichlet Series 120
3 Mellin Transforms 122
4 The Wiener Ikehara Theorem 124
5 Newer Approach to Wiener Ikehara 128
6 Newman's Way to the PNT. Work of Ingham 133
7 Laplace Transforms of Bounded Functions 135
8 Application to Dirichlet Series and the PNT 138
9 Laplace Transforms of Functions Bounded From Below 140
10 Tauberian Conditions Other Than Boundedness 142
11 An Optimal Constant in Theorem 10.1 146
12 Fatou and Riesz. General Dirichlet Series 148
13 Newer Extensions of Fatou Riesz 150
14 Pseudofunction Boundary Behavior 154
15 Applications to Operator Theory 163
16 Complex Remainder Theory 165
17 The Remainder in Fatou's Theorem 167
18 Remainders in Hardy Littlewood Theorems
Involving Power Series 169
19 A Remainder for the Stieltjes Transform 173
Contents XIII
IV Karamata's Heritage: Regular Variation 177
1 Introduction 177
2 Slow and Regular Variation 179
3 Proof of the Basic Properties 181
4 Possible Pathology 184
5 Karamata's Characterization of Regularly Varying Functions 186
6 Related Classes of Functions 188
7 Integral Transforms and Regular Variation: Introduction 191
8 Karamata's Theorem for Laplace Transforms 192
9 Stieltjes and Other Transforms 194
10 The Ratio Theorem 197
11 Beurling Slow Variation 199
12 A Result in Higher Order Theory 200
13 Mercerian Theorems 202
14 Proof of Theorem 13.2 204
15 Asymptotics Involving Large Laplace Transforms 207
16 Transforms of Exponential Growth: Logarithmic Theory 208
17 Strong Asymptotics: General Case 212
18 Application to Exponential Growth 214
19 Very Large Laplace Transforms 217
20 Logarithmic Theory for Very Large Transforms 219
21 Large Transforms: Complex Approach 222
22 Proof of Proposition 21.4 226
23 Asymptotics for Partitions 228
24 Two Sided Laplace Transforms 232
V Extensions of the Classical Theory 235
1 Introduction 235
2 Preliminaries on Banach Algebras 236
3 Algebraic Form of Wiener's Theorem 237
4 Weighted Ll Spaces 239
5 Gelfand's Theory of Maximal Ideals 241
6 Application to the Banach Algebra Aw = (L^, C) 244
7 Regularity Condition for Z,w 246
8 The Closed Maximal Ideals in Lw 248
9 Related Questions Involving Weighted Spaces 250
10 A Boundedness Theorem of Pitt 250
11 Proof of Theorem 10.2, Part 1 252
12 Theorem 10.2: Proof that S(v) = O(ee ) 254
13 Theorem 10.2: Proof that S(y) = 0{e*(v)) 255
14 Boundedness Through Functional Analysis 257
15 Limitable Sequences as Elements of an FA" space 257
16 Perfect Matrix Methods 259
17 Methods with Sectional Convergence 261
18 Existence of (Limitable) Bounded Divergent Sequences 262
XIV Contents
19 Bounded Divergent Sequences, Continued 264
20 Gap Tauberian Theorems 266
21 The Abel Method 268
22 Recurrent Events 270
23 The Theorem of Erdos, Feller and Pollard 272
24 Milin's Theorem 274
25 Some Propositions 275
26 Proof of Milin's Theorem 278
VI Borel Summability and General Circle Methods 279
1 Introduction 279
2 The Methods B and B' 282
3 Borel Summability of Power Series 283
4 The Borel Polygon 285
5 General Circle Methods Tk 288
6 Auxiliary Estimates 290
7 Series with Ostrowski Gaps 292
8 Boundedness Results 295
9 Integral Formulas for Limitability 297
10 Integral Formulas: Case of Positive sn 300
11 First Form of the Tauberian Theorem 302
12 General Tauberian Theorem with Schmidt's Condition 305
13 Tauberian Theorem: Case of Positive sn 306
14 An Application to Number Theory 310
15 High Indices Theorems 311
16 Restricted High Indices Theorem for General Circle Methods 313
17 The Borel High Indices Theorem 317
18 Discussion of the Tauberian Conditions 319
19 Growth of Power Series with Square Root Gaps 323
20 Euler Summability 325
21 The Taylor Method and Other Special Circle Methods 328
22 The Special Methods as Yx Methods 331
23 High Indices Theorems for Special Methods 334
24 Power Series Methods 335
25 Proof of Theorem 24.4 338
VII Tauberian Remainder Theory 343
1 Introduction 343
2 Power Series and Laplace Transforms:
How the Theory Developed 344
3 Theorems for Laplace Transforms 351
4 Proof of Theorems 3.1 and 3.2 354
5 One Sided L' Approximation 356
6 Proof of Proposition 5.2 358
7 Approximation of Smooth Functions 362
Contents XV
8 Proof of Approximation Theorem 3.4 364
9 Vanishing Remainders: Theorem 3.3 366
10 Optimality of the Remainder Estimates 369
11 Dirichlet Series and High Indices 373
12 Proof of Theorem 11.2, Continued 377
13 The Fourier Integral Method: Introduction 379
14 Fourier Integral Method: A Model Theorem 381
15 Auxiliary Inequality of Ganelius 383
16 Proof of the Model Theorem 386
17 A More General Theorem 389
18 Application to Stieltjes Transforms 392
19 Fourier Integral Method: Laplace Stieltjes Transform 394
20 Related Results 398
21 Nonlinear Problems of Erdos for Sequences 400
22 Introduction to the Proof of Theorem 21.3 403
23 Proof of Theorem 21.3, Continued 406
24 An Example and Some Remarks 408
25 Introduction to the Proof of Theorem 21.5 410
26 The Fundamental Relation and a Reduction 412
27 Proof of Theorem 25.1, Continued 415
28 The End Game 417
References 421
Index 469 |
| any_adam_object | 1 |
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| dewey-full | 515.24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV019322132 |
| illustrated | Illustrated |
| indexdate | 2025-03-23T09:00:08Z |
| institution | BVB |
| isbn | 354021058X 9783642059193 |
| language | German |
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| spelling | Korevaar, Jacob 1923-2025 Verfasser (DE-588)10461045X aut Tauberian theory a century of developments Jacob Korevaar Berlin [u.a.] Springer 2004 XV, 483 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften 329 Hier auch später erschienene, unveränderte Nachdrucke Tauber-Sätze Hardy-Littlewood method Summability theory Tauberian theorems Tauber-Sätze (DE-588)4319823-5 gnd rswk-swf Tauber-Sätze (DE-588)4319823-5 s DE-604 Die Grundlehren der mathematischen Wissenschaften 329 (DE-604)BV000000395 329 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012789393&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Korevaar, Jacob 1923-2025 Tauberian theory a century of developments Die Grundlehren der mathematischen Wissenschaften Tauber-Sätze Hardy-Littlewood method Summability theory Tauberian theorems Tauber-Sätze (DE-588)4319823-5 gnd |
| subject_GND | (DE-588)4319823-5 |
| title | Tauberian theory a century of developments |
| title_auth | Tauberian theory a century of developments |
| title_exact_search | Tauberian theory a century of developments |
| title_full | Tauberian theory a century of developments Jacob Korevaar |
| title_fullStr | Tauberian theory a century of developments Jacob Korevaar |
| title_full_unstemmed | Tauberian theory a century of developments Jacob Korevaar |
| title_short | Tauberian theory |
| title_sort | tauberian theory a century of developments |
| title_sub | a century of developments |
| topic | Tauber-Sätze Hardy-Littlewood method Summability theory Tauberian theorems Tauber-Sätze (DE-588)4319823-5 gnd |
| topic_facet | Tauber-Sätze Hardy-Littlewood method Summability theory Tauberian theorems |
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