Overview

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 . . . . . . .

Full Product Details

Author:   Z. Ditzian ,  V. Totik
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1987
Volume:   9
Dimensions:   Width: 15.50cm , Height: 1.20cm , Length: 23.50cm
Weight:   0.371kg
ISBN:  

9781461291510

ISBN 10:   1461291518
Pages:   227
Publication Date:   12 October 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

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.

Reviews

Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions