Moduli of Smoothness

1. Front Matter

   Pages i-ix

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2. Introduction

   1. Introduction

      • Z. Ditzian, V. Totik

      Pages 1-4

3. The Modulus of Smoothness

   1. Front Matter

      Pages 5-5

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   2. Preliminaries

      • Z. Ditzian, V. Totik

      Pages 7-9

   3. The K-Functional and the Modulus of Continuity

      • Z. Ditzian, V. Totik

      Pages 10-23

   4. K-Functionals and Moduli of Smoothness, Other Forms

      • Z. Ditzian, V. Totik

      Pages 24-35

   5. Properties of ω ^r _ϕ (f,t) _p

      • Z. Ditzian, V. Totik

      Pages 36-45

   6. More General Step-Weight Functions ϕ

      • Z. Ditzian, V. Totik

      Pages 46-54

   7. Weighted Moduli of Smoothness

      • Z. Ditzian, V. Totik

      Pages 55-74

4. Applications

   1. Front Matter

      Pages 75-75

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   2. Algebraic Polynomial Approximation

      • Z. Ditzian, V. Totik

      Pages 77-89

   3. Weighted Best Polynomial Approximation

      • Z. Ditzian, V. Totik

      Pages 90-111

   4. Exponential-Type or Bernstein-Type Operators

      • Z. Ditzian, V. Totik

      Pages 112-157

   5. Weighted Approximations by Exponential-Type Operators

      • Z. Ditzian, V. Totik

      Pages 158-179

   6. Weighted Polynomial Approximation in L _p (R)

      • Z. Ditzian, V. Totik

      Pages 180-196

   7. Polynomial Approximation in Several Variables

      • Z. Ditzian, V. Totik

      Pages 197-210

   8. Comparisons and Conclusions

      • Z. Ditzian, V. Totik

      Pages 211-216

5. Back Matter

   Pages 217-227

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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 . . . . . . .

• Approximation
• Interpolation
• approximation theory
• numerical analysis
• real analysis

• Department of Mathematics, University of Alberta, Edmonton, Canada

  Z. Ditzian

• Bolyai Institute, Attila Jozsef University, Szeged, Hungary

  V. Totik

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