Moduli of Smoothness - V. Totik

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Résumé :
The subject of this book is the introduction and application of a new measure for smoothness offunctions. Though we have both previously published some articles in this direction, the results given here are new. Much of the work was done in the summer of 1984 in Edmonton when we consolidated earlier ideas and worked out most of the details of the text. It took another year and a half to improve and polish many of the theorems. We express our gratitude to Paul Nevai and Richard Varga for their encouragement. We thank NSERC of Canada for its valuable support. We also thank Christine Fischer and Laura Heiland for their careful typing of our manuscript. z. Ditzian V. Totik CONTENTS Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 PART I. THE MODULUS OF SMOOTHNESS Chapter 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. Discussion of Some Conditions on cp(x). . . . ? . . . . . . . ? . . ? . . ? ? . 8 . . . ? . 1.3. Examples of Various Step-Weight Functions cp(x) . . ? . . ? . . ? . . ? . . . 9 . . ? Chapter 2. The K-Functional and the Modulus of Continuity ... . ... 10 2.1. The Equivalence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . 2.2. The Upper Estimate, Kr.tp(f, tr)p ~ Mw;(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. K-Functionals and Moduli of Smoothness, Other Forms. 24 3.1. A Modified K-Functional. . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . 3.2. Forward and Backward Differences. . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . 3.3. Main-Part Modulus of Smoothness. . . . . . . . . . . . . . . . . . . . . . 28 . . . . . . .

Sommaire:
I. The Modulus of Smoothness.- 1. Preliminaries.- 2. The K-Functional and the Modulus of Continuity.- 3. K-Functionals and Moduli of Smoothness, Other Forms.- 4. Properties of ??r(f,t)p.- 5. More General Step-Weight Functions ?.- 6. Weighted Moduli of Smoothness.- II. Applications.- 7. Algebraic Polynomial Approximation.- 8. Weighted Best Polynomial Approximation.- 9. Exponential-Type or Bernstein-Type Operators.- 10. Weighted Approximations by Exponential-Type Operators.- 11. Weighted Polynomial Approximation in LP(R).- 12. Polynomial Approximation in Several Variables.- 13. Comparisons and Conclusions.- A. The Analogue of Definition 5.3.1.- B. The Definition of the Weighted Modulus of Smoothness on (0,1).- References.- List of Symbols.