Moduli of Smoothness

  • Z. Ditzian
  • V. Totik

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 9)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Introduction

    1. Z. Ditzian, V. Totik
      Pages 1-4
  3. The Modulus of Smoothness

    1. Front Matter
      Pages 5-5
    2. Z. Ditzian, V. Totik
      Pages 7-9
    3. Z. Ditzian, V. Totik
      Pages 10-23
    4. Z. Ditzian, V. Totik
      Pages 24-35
    5. Z. Ditzian, V. Totik
      Pages 36-45
    6. Z. Ditzian, V. Totik
      Pages 46-54
    7. Z. Ditzian, V. Totik
      Pages 55-74
  4. Applications

    1. Front Matter
      Pages 75-75
    2. Z. Ditzian, V. Totik
      Pages 77-89
    3. Z. Ditzian, V. Totik
      Pages 90-111
    4. Z. Ditzian, V. Totik
      Pages 112-157
    5. Z. Ditzian, V. Totik
      Pages 158-179
    6. Z. Ditzian, V. Totik
      Pages 180-196
    7. Z. Ditzian, V. Totik
      Pages 197-210
    8. Z. Ditzian, V. Totik
      Pages 211-216
  5. Back Matter
    Pages 217-227

About this book

Introduction

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 . . . . . . .

Keywords

Approximation Interpolation approximation theory numerical analysis real analysis

Authors and affiliations

  • Z. Ditzian
    • 1
  • V. Totik
    • 2
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Bolyai InstituteAttila Jozsef UniversitySzegedHungary

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-4778-4
  • Copyright Information Springer-Verlag New York 1987
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-9151-0
  • Online ISBN 978-1-4612-4778-4
  • Series Print ISSN 0179-3632
  • About this book