Exact Constants in Approximation Theory

N. Korneichuk; K. Ivanov

doi:10.1017/CBO9781107325791

This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.

"...a good summary of the Russian literature in the area and would be (especially the bibliography) useful to workers in approximation theory who encounter best constant problems." Terence M. Mills, Mathematical Reviews

"...will be useful to mathematicians who work in approximation theory and are interested in best-constant questions...some people who work in numerical analysis or computational mathematics may also be interested in it." T.M. Mills, The Mathematical Intelligencer

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Exact Constants in Approximation Theory

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-xi

List of notation
pp xii-xii

1 - Best approximation and duality in extremal problems
pp 1-43

2 - Polynomials and spline functions as approximating tools
pp 44-100

3 - Comparison theorems and inequalities for the norms of functions and their derivatives
pp 101-154

4 - Polynomial approximation of classes of functions with bounded rth derivative in Lp
pp 155-210

5 - Spline approximation of classes of functions with a bounded rth derivative
pp 211-259

6 - Exact constants in Jackson inequalities
pp 260-298

7 - Approximation of classes of functions determined by modulus of continuity
pp 299-355

8 - N-widths of functional classes and closely related extremal problems
pp 356-409

Appendix
pp 410-425

References
pp 426-449

Index of notation
pp 450-451

Index
pp 452-452

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