Limit theorems of polynomial approximation with exponential weights:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2008
|
| Series: | Memoirs of the American Mathematical Society
897 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | "Volume 192, number 897 (second of 5 numbers)" Includes bibliographical references and index |
| Physical Description: | VII, 161 S. |
| ISBN: | 9780821840634 |
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| 245 | 1 | 0 | |a Limit theorems of polynomial approximation with exponential weights |c Michael I. Ganzburg |
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| 490 | 1 | |a Memoirs of the American Mathematical Society |v 897 | |
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| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Approximation theory | |
| 650 | 4 | |a Potential theory (Mathematics) | |
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Record in the Search Index
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| adam_text | Contents
Chapter 1. Introduction 1
1.1. A Brief Review 1
1.2. Results and Organization of the Monograph 5
1.3. Basic Notation and Some Preliminaries 7
1.4. Classes of Weights and Basic Estimates 8
1.5. Acknowledgements 13
Chapter 2. Statement of Main Results 15
2.1. Limit Theorems of Polynomial Approximation with Exponential
Weights 15
2.2. Approximation of Entire Functions of Exponential Type 16
2.3. Polynomial Inequalities in the Complex Plane 17
Chapter 3. Properties of Harmonic Functions 19
3.1. The Poisson Integral Re H(w) 19
3.2. The Function h(r) and the Constant bn 24
3.3. The Functions £(r) and fair) 29
3.4. The Main Estimate for Re H(w) 35
Chapter 4. Polynomial Inequalities with Exponential Weights 43
4.1. Nikolskii-type Inequalities 43
4.2. Extremal Polynomials 45
4.3. Polynomial Inequalities in the Complex Plane 55
4.4. Proofs of Theorems 2.3.1 and 2.3.2 57
Chapter 5. Entire Functions of Exponential Type and their Approximation
Properties 59
5.1. Entire Functions of Exponential Type 59
5.2. Approximation Properties of Entire Functions of Exponential Type 62
Chapter 6. Polynomial Interpolation and Approximation of Entire Functions
of Exponential Type 67
6.1. Interpolation on the Interval /„ = [—an(l + 5n), an(l + 6n)] 67
6.2. Interpolation on I /„ 71
6.3. Proof of Theorem 2.2.1 72
6.4. Proof of Theorem 2.2.2 74
Chapter 7. Proofs of the Limit Theorems 77
7.1. Proof of Theorem 2.1.1 77
7.2. Proof of Theorem 2.1.2 80
7.3. Proofs of Theorems 2.1.3 and 2.1.4 82
V
vi CONTENTS
Chapter 8. Applications 85
8.1. Approximation of Individual Functions and Proof of Theorem 2.3.3 85
8.2. An Asymptotically Sharp Constant of Weighted Approximation on
the Class WrHX[I] 96
8.3. Convergence of Polynomials and a Mehler-Heine Formula for
Orthonormal Polynomials 100
Chapter 9. Multidimensional Limit Theorems of Polynomial Approximation
with Exponential Weights 105
9.1. Multidimensional Limit Theorems with Exponential Weights 105
9.2. Proof of Theorem 9.1.3 108
9.3. Proofs of Theorems 9.1.1 and 9.1.4 111
9.4. Approximation of A-Homogeneous Functions 113
Chapter 10. Examples 117
10.1. W(x) = exp(- x a), a 1 117
10.2. W(x) = exp(-ja;j) 121
10.3. W(x) = exp(-|a:|a), 0 a 1 127
10.4. W(x) = exp(-|z|a), a - oo 132
10.5. Examples of Erdos Weights 134
Appendix A. Appendix. Negativity of a Kernel 137
A.I. Statement of the Main Result 137
A.2. Some Technical Results 138
A.3. Proof of Proposition A.I.1 144
Bibliography 155
Index 161
|
| adam_txt |
Contents
Chapter 1. Introduction 1
1.1. A Brief Review 1
1.2. Results and Organization of the Monograph 5
1.3. Basic Notation and Some Preliminaries 7
1.4. Classes of Weights and Basic Estimates 8
1.5. Acknowledgements 13
Chapter 2. Statement of Main Results 15
2.1. Limit Theorems of Polynomial Approximation with Exponential
Weights 15
2.2. Approximation of Entire Functions of Exponential Type 16
2.3. Polynomial Inequalities in the Complex Plane 17
Chapter 3. Properties of Harmonic Functions 19
3.1. The Poisson Integral Re H(w) 19
3.2. The Function h(r) and the Constant bn 24
3.3. The Functions £(r) and fair) 29
3.4. The Main Estimate for Re H(w) 35
Chapter 4. Polynomial Inequalities with Exponential Weights 43
4.1. Nikolskii-type Inequalities 43
4.2. Extremal Polynomials 45
4.3. Polynomial Inequalities in the Complex Plane 55
4.4. Proofs of Theorems 2.3.1 and 2.3.2 57
Chapter 5. Entire Functions of Exponential Type and their Approximation
Properties 59
5.1. Entire Functions of Exponential Type 59
5.2. Approximation Properties of Entire Functions of Exponential Type 62
Chapter 6. Polynomial Interpolation and Approximation of Entire Functions
of Exponential Type 67
6.1. Interpolation on the Interval /„ = [—an(l + 5n), an(l + 6n)] 67
6.2. Interpolation on I \ /„ 71
6.3. Proof of Theorem 2.2.1 72
6.4. Proof of Theorem 2.2.2 74
Chapter 7. Proofs of the Limit Theorems 77
7.1. Proof of Theorem 2.1.1 77
7.2. Proof of Theorem 2.1.2 80
7.3. Proofs of Theorems 2.1.3 and 2.1.4 82
V
vi CONTENTS
Chapter 8. Applications 85
8.1. Approximation of Individual Functions and Proof of Theorem 2.3.3 85
8.2. An Asymptotically Sharp Constant of Weighted Approximation on
the Class WrHX[I] 96
8.3. Convergence of Polynomials and a Mehler-Heine Formula for
Orthonormal Polynomials 100
Chapter 9. Multidimensional Limit Theorems of Polynomial Approximation
with Exponential Weights 105
9.1. Multidimensional Limit Theorems with Exponential Weights 105
9.2. Proof of Theorem 9.1.3 108
9.3. Proofs of Theorems 9.1.1 and 9.1.4 111
9.4. Approximation of A-Homogeneous Functions 113
Chapter 10. Examples 117
10.1. W(x) = exp(-\x\a), a 1 117
10.2. W(x) = exp(-ja;j) 121
10.3. W(x) = exp(-|a:|a), 0 a 1 127
10.4. W(x) = exp(-|z|a), a - oo 132
10.5. Examples of Erdos Weights 134
Appendix A. Appendix. Negativity of a Kernel 137
A.I. Statement of the Main Result 137
A.2. Some Technical Results 138
A.3. Proof of Proposition A.I.1 144
Bibliography 155
Index 161 |
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| spelling | Ganzburg, Michael I. Verfasser aut Limit theorems of polynomial approximation with exponential weights Michael I. Ganzburg Providence, RI American Mathematical Society 2008 VII, 161 S. txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society 897 "Volume 192, number 897 (second of 5 numbers)" Includes bibliographical references and index Functions, Entire Approximation theory Potential theory (Mathematics) Fourier analysis Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf Gewichtete Ungleichung (DE-588)4157274-9 gnd rswk-swf Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s Approximationstheorie (DE-588)4120913-8 s Grenzwertsatz (DE-588)4158163-5 s Gewichtete Ungleichung (DE-588)4157274-9 s DE-604 Memoirs of the American Mathematical Society 897 (DE-604)BV008000141 897 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016400652&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ganzburg, Michael I. Limit theorems of polynomial approximation with exponential weights Memoirs of the American Mathematical Society Functions, Entire Approximation theory Potential theory (Mathematics) Fourier analysis Grenzwertsatz (DE-588)4158163-5 gnd Gewichtete Ungleichung (DE-588)4157274-9 gnd Approximationstheorie (DE-588)4120913-8 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4158163-5 (DE-588)4157274-9 (DE-588)4120913-8 (DE-588)4046939-6 |
| title | Limit theorems of polynomial approximation with exponential weights |
| title_auth | Limit theorems of polynomial approximation with exponential weights |
| title_exact_search | Limit theorems of polynomial approximation with exponential weights |
| title_exact_search_txtP | Limit theorems of polynomial approximation with exponential weights |
| title_full | Limit theorems of polynomial approximation with exponential weights Michael I. Ganzburg |
| title_fullStr | Limit theorems of polynomial approximation with exponential weights Michael I. Ganzburg |
| title_full_unstemmed | Limit theorems of polynomial approximation with exponential weights Michael I. Ganzburg |
| title_short | Limit theorems of polynomial approximation with exponential weights |
| title_sort | limit theorems of polynomial approximation with exponential weights |
| topic | Functions, Entire Approximation theory Potential theory (Mathematics) Fourier analysis Grenzwertsatz (DE-588)4158163-5 gnd Gewichtete Ungleichung (DE-588)4157274-9 gnd Approximationstheorie (DE-588)4120913-8 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Functions, Entire Approximation theory Potential theory (Mathematics) Fourier analysis Grenzwertsatz Gewichtete Ungleichung Approximationstheorie Potenzialtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016400652&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008000141 |
| work_keys_str_mv | AT ganzburgmichaeli limittheoremsofpolynomialapproximationwithexponentialweights |