Exact constants in approximation theory / N. Korneichuk ; translated from the Russian by K. Ivanov.

Format:

Book

Author/Creator:

Korneĭchuk, N. P. (Nikolaĭ Pavlovich), author.

Contributor:

Ivanov, K., translator.

Series:

Encyclopedia of mathematics and its applications ; v. 38.

Encyclopedia of mathematics and its applications ; volume 38

Standardized Title:

Tochnye konstanty v teorii priblizhenii︠a︡. English

Language:

English

Subjects (All):

Approximation theory.

Physical Description:

1 online resource (xii, 452 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1990.

Language Note:

English

Summary:

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.

Contents:

Cover; Half Title; Series Page; Title; Copyright; CONTENTS; PREFACE; LIST OF MOST IMPORTANT NOTATION; 1 Best approximation and duality in extremal problems; 1.1 Best approximation; 1.2 Formulation of extremal problems; 1.3 Duality of extremal problems in linear spaces; 1.4 Duality in functional spaces; 1.5 Duality for best approximation of classes of functions; Comments; Exercises; 2 Polynomials and spline functions as approximating tools; 2.1 Polynomials of best approximation; 2.2 Linear methods for polynomial approximation, Lebesgue constants; 2.3 Polynomial splines

2.4 Spline interpolation2.5 On the existence of perfect splines with prescribed zeros; Comments; Exercises; 3 Comparison theorems and inequalities for the norms of functions and their derivatives; 3.1 Standard splines; 3.2 Comparison theorems in general cases; 3.3 Comparison theorems and exact inequalities for differentiable functions; 3.4 Internal extremal properties of splines; 3.5 Inequalities for polynomials; Comments; Other results and exercises; 4 Polynomial approximation of classes of functions with bounded rth derivative in Lp; 4.1 Minimizing the error in the class of A-methods

4.2 The supremums of the best approximations of classes Wrp by trigonometric polynomials4.3 Approximation by partial sums of Fourier series, their means and analogs; 4.4 Approximation by algebraic polynomials on an interval; Comments; Other results and exercises; 5 Spline approximation of classes of functions with a bounded rth derivative; 5.1 Inequalities for functions with prescribed zeros; 5.2 The interpolation error of the periodic splines with minimal defect on classes Wrp; 5.3 Estimates for spline interpolation on classes Wrp[a,b]; 5.4 Best approximation by splines of minimal defect

CommentsOther results and exercises; 6 Exact constants in Jackson inequalities; 6.1 Modulus of continuity and general statement of the problems; 6.2 Jackson inequalities for polynomial approximations; 6.3 Jackson-type inequalities for spline approximation; Comments; Other results and exercises; 7 Approximation of classes of functions determined by modulus of continuity; 7.1 General facts and Σ-rearrangements; 7.2 Exact results for the best approximation of classes WrHw, WrHw[a,b]; 7.3 Approximation by linear methods; Comments; Other results and exercises

8 N-widths of functional classes and closely related extremal problems8.1 N-widths of classes of functions with bounded rth derivatives; 8.2 N-widths of classes defined by the modulus of continuity; 8.3 Closely related extremal problems; Comments; Other results and exercises; APPENDIX; A1 Hölder and Minkowski inequalities and some extremal relations; A2 Steklov functions; A3 Proofs of comparison theorems for Σ-rearrangements; 1 Auxiliary results; 2 Proofs of Theorems 7.1.4 and 7.1.5; 3 Proof of theorem 7.1.6; REFERENCES; I Monographs; A Monographs on approximation theory; B Other books

II Articles

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 426-449) and indexes.

ISBN:

1-139-88632-0

1-107-10273-1

1-107-08811-9

1-107-10017-8

1-107-09427-5

1-107-32579-X