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Geometric approximation theory

Alimov, Alexey R.

Geometric approximation theory

Record Type:	Electronic resources : Monograph/item
Title/Author:	Geometric approximation theoryby Alexey R. Alimov, Igor' G. Tsar'kov.
Author:	Alimov, Alexey R.
other author:	Tsar'kov, Igor' G.
Published:	Cham :Springer International Publishing :2021.
Description:	1 online resource (xxi, 508 p.) :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Approximation theory.
Online resource:	https://doi.org/10.1007/978-3-030-90951-2
ISBN:	9783030909512$q(electronic bk.)

Geometric approximation theory
Alimov, Alexey R.

Geometric approximation theory[electronic resource] /by Alexey R. Alimov, Igor' G. Tsar'kov. - Cham :Springer International Publishing :2021. - 1 online resource (xxi, 508 p.) :ill., digital ;24 cm. - Springer monographs in mathematics,2196-9922. - Springer monographs in mathematics..

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.

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. It concludes 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.

ISBN: 9783030909512$q(electronic bk.)

Standard No.: 10.1007/978-3-030-90951-2doiSubjects--Topical Terms:

185327
Approximation theory.

LC Class. No.: QA221 / A55 2021

Dewey Class. No.: 511.4

Geometric approximation theory
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505 0 $a 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.
520 $a 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. It concludes 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.
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