Exact Constants in Approximation Theory

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Paperback | 11 June 2009

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468 Pages

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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.

Industry Reviews

"...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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Preface
Best approximation and duality in extremal problems
Polynomials and spline-functions as approximating tools
Comparison theorems and inequalities for the norms of functions and their derivatives
Polynomial approximation of classes of functions with bounded r-th derivative in Lp
Spline approximation of classes of functions with bounded r-th derivative
Exact constants in Jackson inequalities
Approximation of classes of functions determined by modulus of continuity
N-widths of functional classes and closely related extremal problems
Appendixes
References
Table of Contents provided by Publisher. All Rights Reserved.

Exact Constants in Approximation Theory

At a Glance

Paperback

Industry Reviews

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