Alimov A.R., Tsar'kov I.G. Geometric Approximation Theory

Springer, 2022. — 522 p. — (Springer Monographs in Mathematics). — ISBN 3030909506.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.Preeface.
Main Notation, Definitions, Auxiliary Results, and Examples.
Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems.
Best Approximation in Euclidean Spaces.
Existence. Compact, Boundedly Compact, Approximatively Compact, and s-Compact Sets. Continuity of the Metric Projection.
Characterization of Best Approximation and Solar Properties of Sets.
Convexity of Chebyshev Sets and Suns.
Connectedness and Approximative Properties of Sets. Stability of the Metric Projection and Its Relation to Other Approximative Properties.
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 Varisolvency.
Approximation of Vector-Valued Functions.
The Jung Constant.
Chebyshev Centre of a Set. The Problem of Simultaneous Approximation of a Class by a Singleton Set.
Width. Approximation by a Family of Sets.
Approximative Properties of Arbitrary Sets in Normed Linear Spaces. Almost Chebyshev Sets and Sets of Almost Uniqueness.
A Chebyshev Systems of Functions in the Spaces C , Cn and Lp.
B Radon, Helly, and Carathéodory Theorems. Decomposition Theorem.
C Some Open Problems.True PDF