Moduli of smoothness:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1987
|
| Series: | Springer series in computational mathematics
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 225 S. |
| ISBN: | 038796536X 354096536X |
Staff View
MARC
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| 245 | 1 | 0 | |a Moduli of smoothness |c Z. Ditzian ; V. Totik |
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| 650 | 4 | |a Fonctions de lissage | |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Introduction 1
PART I. THE MODULUS OF SMOOTHNESS
Chapter 1. Preliminaries 7
1.1. Notations 7
1.2. Discussion of Some Conditions on ip(x) 8
1.3. Examples of Various Step Weight Functions p{x) 9
Chapter 2. The K Functional and the Modulus of Continuity 10
2.1. The Equivalence Theorem 10
2.2. The Upper Estimate, *,,„(/, t% Mco v(f, t)p, Case I 12
2.3. The Upper Estimate of the K Functional, The Other Cases 16
2.4. The Lower Estimate for the K Functional 20
Chapter 3. X Functionals and Moduli of Smoothness, Other Forms . 24
3.1. A Modified ^ Functional 24
3.2. Forward and Backward Differences 26
3.3. Main Part Modulus of Smoothness 28
3.4. Computation of Our Modulus for Some Functions 34
Chapter 4. Properties of corQ(f, t)p 36
4.1. Extending the Basic Properties of the Classical Moduli 36
4.2. Optimal Rate of co v(f, t) 40
4.3. Marchaud Inequality 43
viii Contents
Chapter 5. More General Step Weight Functions cp 46
5.1. Logarithmic Type Weights and Internal Zeros 46
5.2. The Necessity of the Finite Overlapping Condition 47
5.3. Growth Order of Type x with Arbitrary fl 49
Chapter 6. Weighted Moduli of Smoothness 55
6.1. Weighted Moduli of Smoothness and Weighted K Functionals 55
6.2. The Weighted Main Part Modulus 59
6.3. Smoothness Properties of Derivatives 62
6.4. Marchaud Inequality for Weighted Main Part Moduli 67
6.5. Connection with Ordinary Weighted Moduli 71
PART II. APPLICATIONS
Chapter 7. Algebraic Polynomial Approximation 77
7.1. Background 77
7.2. Best Polynomial Approximation 79
7.3. Asymptotic Behavior of Derivatives of Best Approximating Polynomials. 84
7.4. Error Bounds for Gaussian Quadrature 87
Chapter 8. Weighted Best Polynomial Approximation 90
8.1. Some Concepts and Description of the Weight 90
8.2. Best Weighted Algebraic Polynomial Approximation 94
8.3. Derivatives of the Optimal Polynomials 98
8.4. Proof of Some Crucial Inequalities for we J* 100
8.5. Applications, Calculations, and Specific Examples 109
Chapter 9. Exponential Type or Bernstein Type Operators 112
9.1. Background and Notations, Positive Operators on C(D) 112
9.2. Operators on LP(D), Higher Degree of Smoothness 115
9.3. Direct and Converse Results 117
9.4. The Bernstein Type Inequality (p l L^r)f p Mnr||/||p 124
9.5. Rate of Convergence for Smooth Functions 134
9.6. Estimate of Ln(R2r(f, ,x),x) L (£ , 140
9.7. The Estimate |! p(x)2rL 2 H/)lli/ M|| p2 / 2 ||p 152
Chapter 10. Weighted Approximations by Exponential Type Operators 158
10.1. The Direct and Inverse Result 158
10.2. The Boundedness of the Operators in Weighted Norm 161
10.3. Bernstein Type Inequality 165
10.4. The Estimate ||w p2Li2)(/)|| C(||w p2/(2)|| + ||/||) 168
10.5. The Estimate of LJ /for Smooth Functions 168
10.6. The Saturation Result 175
Contents ix
Chapter 11. Weighted Polynomial Approximation in Lp(R) 180
11.1. Introduction 180
11.2. The Equivalence Result 181
11.3. The Direct and Converse Results 184
11.4. Proof of the Equivalence Result 186
11.5. Comparisons and Generalizations 195
Chapter 12. Polynomial Approximation in Several Variables 197
12.1. Approximation on Cubes 197
12.2. Approximation on Polytopes 201
Chapter 13. Comparisons and Conclusions 211
13.1. Comparison with Similar Expressions 211
13.2. The Integral Modulus of Smoothness of Ivanov and Sendov 212
13.3. Moduli Generated by Multipliers and Integral Transforms 213
13.4. A Modulus Introduced by Potapov 214
13.5. Hoeffding s Result 215
13.6. Conclusion 216
Appendix 217
A. The Analogue of Definition 5.3.1 217
B. The Definition of the Weighted Modulus of Smoothness on (0,1) 218
References 219
List of Symbols 226
Index 227
|
| any_adam_object | 1 |
| author | Ditzian, Zeev Totik, Vilmos 1954- |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV000788979 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:19:28Z |
| institution | BVB |
| isbn | 038796536X 354096536X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000494045 |
| oclc_num | 16003527 |
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| physical | IX, 225 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Ditzian, Zeev Verfasser aut Moduli of smoothness Z. Ditzian ; V. Totik New York u.a. Springer 1987 IX, 225 S. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 9 Fonctions de lissage Modules, Théorie des Modules, Théorie des ram approximation polynomiale lissage lissage fonction théorie module Moduli theory Smoothness of functions Glättungsmodul (DE-588)4314206-0 gnd rswk-swf Stetigkeitsmodul (DE-588)4183168-8 gnd rswk-swf Stetigkeitsmodul (DE-588)4183168-8 s DE-604 Glättungsmodul (DE-588)4314206-0 s Totik, Vilmos 1954- Verfasser (DE-588)111480485 aut Springer series in computational mathematics 9 (DE-604)BV000012004 9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000494045&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ditzian, Zeev Totik, Vilmos 1954- Moduli of smoothness Springer series in computational mathematics Fonctions de lissage Modules, Théorie des Modules, Théorie des ram approximation polynomiale lissage lissage fonction théorie module Moduli theory Smoothness of functions Glättungsmodul (DE-588)4314206-0 gnd Stetigkeitsmodul (DE-588)4183168-8 gnd |
| subject_GND | (DE-588)4314206-0 (DE-588)4183168-8 |
| title | Moduli of smoothness |
| title_auth | Moduli of smoothness |
| title_exact_search | Moduli of smoothness |
| title_full | Moduli of smoothness Z. Ditzian ; V. Totik |
| title_fullStr | Moduli of smoothness Z. Ditzian ; V. Totik |
| title_full_unstemmed | Moduli of smoothness Z. Ditzian ; V. Totik |
| title_short | Moduli of smoothness |
| title_sort | moduli of smoothness |
| topic | Fonctions de lissage Modules, Théorie des Modules, Théorie des ram approximation polynomiale lissage lissage fonction théorie module Moduli theory Smoothness of functions Glättungsmodul (DE-588)4314206-0 gnd Stetigkeitsmodul (DE-588)4183168-8 gnd |
| topic_facet | Fonctions de lissage Modules, Théorie des approximation polynomiale lissage lissage fonction théorie module Moduli theory Smoothness of functions Glättungsmodul Stetigkeitsmodul |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000494045&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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