Fourier analysis and approximation: 1 One-dimensional theory
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Stuttgart
Birkhäuser
1971
|
| Series: | Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften
Mathematische Reihe ; 40 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 553 S. |
| ISBN: | 3764305207 |
Staff View
MARC
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| 490 | 1 | |a Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften : Mathematische Reihe |v 40 | |
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Record in the Search Index
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| adam_text | Contents
0 Preliminaries
0.1 Fundamentals on Lebesgue Integration 1
0.2 Convolutions on the Line Group 4
0.3 Further Sets of Functions and Sequences 6
0.4 Periodic Functions and Their Convolution 8
0.5 Functions of Bounded Variation on the Line Group 10
0.6 The Class BV2jI 14
0.7 Normed Linear Spaces, Bounded Linear Operators 15
0.8 Bounded Linear Functionals, Riesz Representation Theorems ... 20
0.9 References 24
Parti
Approximation by Singular Integrals 25
1 Singular Integrals of Periodic Functions
1.0 Introduction 29
1.1 Norm Convergence and Derivatives 30
1.1.1 Norm Convergence 30
1.1.2 Derivatives 33
1.2 Summation of Fourier Series 39
1.2.1 Definitions 39
1.2.2 Dirichlet and Fejer Kernel 42
1.2.3 Weierstrass Approximation Theorem 44
1.2.4 Summability of Fourier Series 44
1.2.5 Row Finite 0 Factors 47
1.2.6 Summability of Conjugate Series 47
1.2.7 Fourier Stieltjes Series 49
1.3 Test Sets for Norm Convergence 54
1.3.1 Norms of Some Convolution Operators 54
1.3.2 Some Applications of the Theorem of Banach Steinhaus 55
1.3.3 Positive Kernels 58
1.4 Pointwise Convergence 61
1.5 Order of Approximation for Positive Singular Integrals .... 67
1.5.1 Modulus of Continuity and Lipschitz Classes 67
1.5.2 Direct Approximation Theorems 68
1.5.3 Method of Test Functions 70
1.5.4 Asymptotic Properties 72
Xii CONTENTS
1.6 Further Direct Approximation Theorems, Nikolskil Constants . . 79
1.6.1 Singular Integral of Fejer Korovkin 79
1.6.2 Further Direct Approximation Theorems 80
1.6.3 Nikolskil Constants 82
1.7 Simple Inverse Approximation Theorems 86
1.8 Notes and Remarks 89
2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation
and for Singular Integrals
2.0 Introduction 94
2.1 Polynomials of Best Approximation 95
2.2 Theorems of Jackson 97
2.3 Theorems of Bernstein 99
2.4 Various Applications 104
2.5 Approximation Theorems for Singular Integrals 109
2.5.1 Singular Integral of Abel Poisson 109
2.5.2 Singular Integral of de La Vallee Poussin 112
2.6 Notes and Remarks 116
3 Singular Integrals on the Line Group
3.0 Introduction 119
3.1 Norm Convergence 120
3.1.1 Definitions and Fundamental Properties 120
3.1.2 Singular Integral of Fejer 122
3.1.3 Singular Integral of Gauss Weierstrass 125
3.1.4 Singular Integral of Cauchy Poisson 126
3.2 Pointwise Convergence 132
3.3 Order of Approximation 136
3.4 Further Direct Approximation Theorems 142
3.5 Inverse Approximation Theorems 146
3.6 Shape Preserving Properties 150
3.6.1 Singular Integral of Gauss Weierstrass 150
3.6.2 Variation Diminishing Kernels 154
3.7 Notes and Remarks 158
Part II
Fourier Transforms 163
4 Finite Fourier Transforms
4.0 Introduction 167
4.1 LL Theory 167
4.1.1 Fundamental Properties 167
4.1.2 Inversion Theory 171
4.1.3 Fourier Transforms of Derivatives 172
4.2 L^ Theory, p 1 174
4.2.1 The Case p = 2 174
4.2.2 The Case p * 2 177
CONTENTS xiii
4.3 Finite Fourier Stieltjes Transforms 179
4.3.1 Fundamental Properties 179
4.3.2 Inversion Theory 182
4.3.3 Fourier Stieltjes Transforms of Derivatives 183
4.4 Notes and Remarks 185
5 Fourier Transforms Associated with the Line Group
5.0 Introduction 188
5.1 L^Theory 188
5.1.1 Fundamental Properties 188
5.1.2 Inversion Theory 190
5.1.3 Fourier Transforms of Derivatives 194
5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions,
Peano and Riemann Derivatives 196
5.1.5 Poisson Summation Formula 201
5.2 L Theory, 1 p 2 208
5.2.1 TheCase/ = 2 208
5.2.2 The Case 1 p 2 209
5.2.3 Fundamental Properties 212
5.2.4 Summation of the Fourier Inversion Integral 213
5.2.5 Fourier Transforms of Derivatives 214
5.2.6 Theorem of Plancherel 216
5.3 Fourier Stieltjes Transforms 219
5.3.1 Fundamental Properties 219
5.3.2 Inversion Theory 222
5.3.3 Fourier Stieltjes Transforms of Derivatives 224
5.4 Notes and Remarks 227
6 Representation Theorems
6.0 Introduction 231
6.1 Necessary and Sufficient Conditions 232
6.1.1 Representation of Sequences as Finite Fourier or Fourier Stieltjes Trans¬
forms 232
6.1.2 Representation of Functions as Fourier or Fourier Stieltjes Transforms 235
6.2 Theorems of Bochner 241
6.3 Sufficient Conditions 246
6.3.1 Quasi Convexity 246
6.3.2 Representation as LL Transform 249
6.3.3 Representation as L1 Transform 250
6.3.4 A Reduction Theorem 252
6.4 Applications to Singular Integrals 256
6.4.1 General Singular Integral of Weierstrass 257
6.4.2 Typical Means 261
6.5 Multipliers 266
6.5.1 Multipliers of Classes of Periodic Functions 266
6.5.2 Multipliers on L 268
6.6 Notes and Remarks 273
7 Fourier Transform Methods and Second Order Partial Differential Equations
7.0 Introduction 278
Xiv CONTENTS
7.1 Finite Fourier Transform Method 281
7.1.1 Solution of Heat Conduction Problems 281
7.1.2 Dirichlet s and Neumann s Problem for the Unit Disc 284
7.1.3 Vibrating String Problems 287
7.2 Fourier Transform Method in L1 294
7.2.1 Diffusion on an Infinite Rod 294
7.2.2 Dirichlet s Problem for the Half Plane 297
7.2.3 Motion of an Infinite String 298
7.3 Notes and Remarks 300
Part III
Hilbert Transforms 303
8 Hilbert Transforms on the Real Line
8.0 Introduction 305
8.1 Existence of the Transform 307
8.1.1 Existence Almost Everywhere 307
8.1.2 Existence in La Norm 310
8.1.3 Existence in L Norm, 1 p oo 312
8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert
Transforms 316
8.2.1 Hilbert Formulae 316
8.2.2 Conjugates of Singular Integrals: 1 p oo 318
8.2.3 Conjugates of Singular Integrals: p = 1 320
8.2.4 Iterated Hilbert Transforms 323
8.3 Fourier Transforms of Hilbert Transforms 324
8.3.1 Signum Rule 324
8.3.2 Summation of Allied Integrals 325
8.3.3 Fourier Transforms of Derivatives of Hilbert Transforms, the Classes
(W~)[p, (V~)| 327
8.3.4 Norm Convergence of the Fourier Inversion Integral 329
8.4 Notes and Remarks 331
9 Hilbert Transforms of Periodic Functions
9.0 Introduction 334
9.1 Existence and Basic Properties 335
9.1.1 Existence 335
9.1.2 Hilbert Formulae 338
9.2 Conjugates of Singular Integrals 341
9.2.1 The Case 1 p oo 341
9.2.2 Convergence in C2n and LJ« 341
9.3 Fourier Transforms of Hilbert Transforms 347
9.3.1 Conjugate Fourier Series 347
9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes
(W X,,, (V~fcte 349
9.3.3 Norm Convergence of Fourier Series 350
9.4 Notes and Remarks 353
CONTENTS XV
Part IV
Characterization of Certain Function Classes 355
10 Characterization in the Integral Case
10.0 Introduction 357
10.1 Generalized Derivatives, Characterization of the Classes W£2ji . . . 358
10.1.1 Riemann Derivatives in X2;l Norm 358
10.1.2 Strong Peano Derivatives 361
10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives 363
10.2 Characterization of the Classes VJ2ji 366
10.3 Characterization of the Classes (V~)*2n 371
10.4 Relative Completion 373
10.5 Generalized Derivatives in Lp Norm and Characterizations for
1 p 2 376
10.6 Generalized Derivatives in X((R) Norm and Characterizations of the
Classes WJ(R) and VJ(H) 382
10.7 Notes and Remarks 389
11 Characterization in the Fractional Case
11.0 Introduction 391
11.1 Integrals of Fractional Order 393
11.1.1 Integral of Riemann Liouville 393
11.1.2 Integral of M. Riesz 396
11.2 Characterizations of the Classes W[L ; v a], V[L ; v a], 1 p 2 . 400
11.2.1 Derivatives of Fractional Order 400
11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[L ; v 405
11.3 The Operators Rla} on Lp, 1 p 2 409
11.3.1 Characterizations 409
11.3.2 Theorems of Bernstein Titchmarsh and H. Weyl 414
11.4 The Operators ./? « on X2)l 416
11.5 Integral Representations, Fractional Derivatives of Periodic Functions 419
11.6 Notes and Remarks 428
Part V
Saturation Theory 431
12 Saturation for Singular Integrals on X2jI and L , 1 p 2
12.0 Introduction 433
12.1 Saturation for Periodic Singular Integrals, Inverse Theorems . . . 435
12.2 Favard Classes 440
12.2.1 Positive Kernels 440
12.2.2 Uniformly Bounded Multipliers 441
12.2.3 Functional Equations 446
12.3 Saturation in L , 1 p 2 452
12.3.1 Saturation Property 452
12.3.2 Characterizations of Favard Classes: p = 1 455
12.3.3 Characterizations of Favard Classes: 1 p 2 459
XVi CONTENTS
12.4 Applications to Various Singular Integrals 463
12.4.1 Singular Integral of Fejer 463
12.4.2 Generalized Singular Integral of Picard 464
12.4.3 General Singular Integral of Weierstrass 465
12.4.4 Singular Integral of Bochner Riesz 467
12.4.5 Riesz Means 469
12.5 Saturation of Higher Order 471
12.5.1 Singular Integrals on the Real Line 471
12.5.2 Periodic Singular Integrals 474
12.6 Notes and Remarks 478
13 Saturation on X(R)
13.0 Introduction 483
13.1 Saturation of Dff(/;x;r) in X(R), Dual Methods 485
13.2 Applications to Approximation in Lp, 2 p oo 488
13.2.1 Differences 488
13.2.2 Singular Integrals Satisfying (12.3.5) 489
13.2.3 Strong Riesz Derivatives 490
13.2.4 The Operators R^ 491
13.2.5 Riesz and Fejer Means 492
13.3 Comparison Theorems 493
13.3.1 Global Divisibility 493
13.3.2 Local Divisibility 495
13.3.3 Special Comparison Theorems with no Divisibility Hypothesis 498
13.3.4 Applications to Periodic Continuous Functions 500
13.4 Saturation on Banach Spaces 502
13.4.1 Strong Approximation Processes 502
13.4.2 Semi Groups of Operators 504
13.5 Notes and Remarks 507
List of Symbols 511
Tables of Fourier and Hilbert Transforms 515
Bibliography 521
Index 547
|
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| bvnumber | BV009384408 |
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| discipline | Mathematik |
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| id | DE-604.BV009384408 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:35:22Z |
| institution | BVB |
| isbn | 3764305207 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006227044 |
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| physical | XVI, 553 S. |
| publishDate | 1971 |
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| series | Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften |
| series2 | Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften : Mathematische Reihe |
| spelling | Butzer, Paul L. 1928- Verfasser (DE-588)115500529 aut Fourier analysis and approximation 1 One-dimensional theory Paul L. Butzer ; Rolf J. Nessel Stuttgart Birkhäuser 1971 XVI, 553 S. txt rdacontent n rdamedia nc rdacarrier Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften : Mathematische Reihe 40 Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften : Mathematische Reihe ... Approximation (DE-588)4002498-2 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Integraltransformation (DE-588)4027235-7 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s Integraltransformation (DE-588)4027235-7 s Approximation (DE-588)4002498-2 s DE-604 Nessel, Rolf Joachim Verfasser aut (DE-604)BV003801790 1 Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften Mathematische Reihe ; 40 (DE-604)BV000003214 40 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006227044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Butzer, Paul L. 1928- Nessel, Rolf Joachim Fourier analysis and approximation Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften Approximation (DE-588)4002498-2 gnd Harmonische Analyse (DE-588)4023453-8 gnd Integraltransformation (DE-588)4027235-7 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4023453-8 (DE-588)4027235-7 |
| title | Fourier analysis and approximation |
| title_auth | Fourier analysis and approximation |
| title_exact_search | Fourier analysis and approximation |
| title_full | Fourier analysis and approximation 1 One-dimensional theory Paul L. Butzer ; Rolf J. Nessel |
| title_fullStr | Fourier analysis and approximation 1 One-dimensional theory Paul L. Butzer ; Rolf J. Nessel |
| title_full_unstemmed | Fourier analysis and approximation 1 One-dimensional theory Paul L. Butzer ; Rolf J. Nessel |
| title_short | Fourier analysis and approximation |
| title_sort | fourier analysis and approximation one dimensional theory |
| topic | Approximation (DE-588)4002498-2 gnd Harmonische Analyse (DE-588)4023453-8 gnd Integraltransformation (DE-588)4027235-7 gnd |
| topic_facet | Approximation Harmonische Analyse Integraltransformation |
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