Geometric approximation theory / Alexey R. Alimov, Igorʹ G. Tsarʹkov.

Format:

Book

Author/Creator:

Alimov, Alexey, author.

Tsar'kov, I. G. (Igor' G.), author.

Series:

Springer monographs in mathematics 1439-7382

Springer monographs in mathematics, 1439-7382

Language:

English

Subjects (All):

Approximation theory.

Geometry.

Chebyshev systems.

geometry.

Physical Description:

xxi, 508 pages ; 25 cm.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

"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"--Back cover.

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.

Notes:

Includes bibliographical references and index.

ISBN:

9783030909505

3030909506

OCLC:

1319809892