Exact constants in approximation theory / N. Korneichuk ; translated from the Russian by K. Ivanov.
1991
QA221
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Details
Title
Exact constants in approximation theory / N. Korneichuk ; translated from the Russian by K. Ivanov.
Uniform Title
Tochnye konstanty v teorii priblizhenii͡a. English
ISBN
9781107088115 (electronic bk.)
1107088119 (electronic bk.)
9781107325791 (electronic bk.)
110732579X (electronic bk.)
0521382343
9780521382342
9780521111560
0521111560
1107088119 (electronic bk.)
9781107325791 (electronic bk.)
110732579X (electronic bk.)
0521382343
9780521382342
9780521111560
0521111560
Imprint
Cambridge ; New York : Cambridge University Press, 1991.
Language
English
Language Note
Translation of: Tochnye konstanty v teorii priblizhenii͡.
Description
1 online resource (xii, 452 pages) : illustrations
Call Number
QA221
System Control No.
(OCoLC)852896262
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.
Note
Translation of: Tochnye konstanty v teorii priblizhenii͡a.
Bibliography, etc. Note
Includes bibliographical references (pages 426-449) and index.
Formatted Contents Note
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
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
Source of Description
Print version record.
Series
Encyclopedia of mathematics and its applications.
Available in Other Form
Print version: Korneĭchuk, Nikolaĭ Pavlovich. Tochnye konstanty v teorii priblizhenii͡a. English. Exact constants in approximation theory. Cambridge ; New York : Cambridge University Press, 1991
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