Overview

This monograph provides a comprehensive introduction to the classical geometric approximation theory, emphasizing important themes related to the theory including uniqueness, stability, and existence of elements of best approximation. It presents a number of fundamental results for both these and related problems, many of which appear for the first time in monograph form. The text also discusses the interrelations between main objects of geometric approximation theory, formulating a number of auxiliary problems for demonstration. Central ideas include the problems of existence and uniqueness of elements of best approximations as well as properties of sets including subspaces of polynomials and splines, classes of rational functions, and abstract subsets of normed linear spaces. The book begins with a brief introduction to geometric approximation theory, progressing through fundamental classical ideas and results as a basis for various approximation sets, suns, and Chebyshev systems. Itconcludes with a review of approximation by abstract sets and related problems, presenting novel results throughout the section. This text is suitable for both theoretical and applied viewpoints and especially researchers interested in advanced aspects of the field. 

Full Product Details

Author:   Alexey R. Alimov ,  Igor’ G. Tsar’kov
Publisher:   Springer Nature Switzerland AG
Imprint:   Springer Nature Switzerland AG
Edition:   1st ed. 2021
Weight:   0.807kg
ISBN:  

9783030909536

ISBN 10:   3030909530
Pages:   508
Publication Date:   30 March 2023
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

Main notation, definitions, auxillary results, and examples.- Chebyshev alternation theorem, Haar and Mairhuber's theorems.- Best approximation in Euclidean spaces.- Existence and compactness.- Characterization of best approximation.- Convexity of Chebyshev sets and sums.- Connectedness and stability.- Existence of Chebyshev subspaces.- Efimov–Stechkin spaces. Uniform convexity and uniform smoothness. Uniqueness and strong uniqueness of best approximation in uniformly convex spaces.- Solarity of Chebyshev sets.- Rational approximation.- Haar cones and varisolvencity.- Approximation of vector-valued functions.- The Jung constant.- Chebyshev centre of a set.- Width. Approximation by a family of sets.- Approximative properties of arbitrary sets.- Chebyshev systems of functions in the spaces C, Cn, and Lp.- Radon, Helly, and Carathéodory theorems. Decomposition theorem.- Some open problems.- Index.

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