Geometric approximation theory / Alexey R. Alimov and Igor' G. Tsar'kov.

Format:

Book

Author/Creator:

Alimov, Alexey, author.

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

Series:

Springer Monographs in Mathematics

Language:

English

Subjects (All):

Approximation theory.

Approximation theory--Data processing.

Physical Description:

1 online resource (523 pages)

Edition:

1st ed.

Place of Publication:

Cham, Switzerland : Springer, [2022]

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.

Contents:

Intro

Preface

Contents

1 Main Notation, Definitions, Auxiliary Results, and Examples

1.1 Main Definitions of Geometric Approximation Theory

1.2 Preliminaries and Some Facts from Functional Analysis

1.3 Elementary Results on Best Approximation. Strictly Convex Spaces. Approximation by Subspaces and Hyperplanes

2 Chebyshev Alternation Theorem. Haar's and Mairhuber's Theorems

2.1 Chebyshev's and de la Vallée Poussin's Theorems

2.2 Solarity and Alternant

2.3 Haar's Theorem. Strong Uniqueness of Best Approximation

2.4 A Short Note on Extremal Signatures

2.5 Mairhuber's Theorem

2.6 Approximation of Continuous Functions by Finite-Dimensional Subspaces in the L1-Metric

2.7 Remez's Algorithm for Construction of a Polynomials of Near-Best Approximation

3 Best Approximation in Euclidean Spaces

3.1 Approximation by Convex Sets. Kolmogorov Criterion for a Nearest Element. Deutsch's Lemma

3.2 Phelps's Theorem on the Lipschitz Continuity of the Metric Projection onto Chebyshev Sets

3.3 Best Least-Squares Polynomial Approximation. Orthogonal Polynomials

4 Existence. Compact, Boundedly Compact, Approximatively Compact, and τ-Compact Sets. Continuity of the Metric Projection

4.1 Boundedly Compact and Approximatively Compact Sets

4.2 Existence of Best Approximation

4.3 Approximative τ-Compactness with Respect to Regular τ-Convergence

4.3.1 Applications in C[a,b]

4.3.2 Applications in Lp

5 Characterization of Best Approximation and Solar Properties of Sets

5.1 Characterization of an Element of Best Approximation

5.2 Suns and the Kolmogorov Criterion for a Nearest Element. Local and Global Best Approximation. Unimodal Sets (LG-Sets)

5.3 Kolmogorov Criterion in the Space C(Q)

5.4 Continuity of the Metric Projection onto Chebyshev Sets.

5.5 Differentiability of the Distance Function

5.6 Relation of Geometric Approximation Theory to Geometric Optics

6 Convexity of Chebyshev Sets and Suns

6.1 Convexity of Suns

6.2 Convexity of Chebyshev Sets in mathbbRn

6.2.1 Berdyshev-Klee-Vlasov's proof

6.2.2 Asplund's Proof

6.2.3 Konyagin's Proof

6.2.4 Vlasov's Proof

6.2.5 Brosowski's Proof

6.3 The Klee Cavern

6.4 Johnson's Example of a Nonconvex Chebyshev Set in an Incomplete Pre-Hilbert Space

7 Connectedness and Approximative Properties of Sets. Stability of the Metric Projection and Its Relation to Other Approximative Properties

7.1 Classes of Connectedness of Sets

7.2 Connectedness of Suns

7.3 Dunham's Example of a Disconnected Chebyshev Set with Isolated Point

7.4 Klee's Example of a Discrete Chebyshev Set

7.5 Koshcheev's Example of a Disconnected Sun

7.6 Radial Continuity of the Metric Projection. B-Connectedness of Approximatively Compact Chebyshev Suns

7.7 Spans, Segments. Menger Connectedness, and Monotone Path-Connectedness

7.7.1 The Banach-Mazur Hull

7.7.2 Segments and Spans in Normed Linear Spaces

7.7.3 Monotone Path-Connectedness

7.8 Continuous and Semicontinuous Selections of Metric Projection. Relation to Solarity and Proximinality of Sets

7.9 Suns, Unimodal Sets, Moons, and ORL-Continuity. Brosowski-Wegmann-connectedness

7.10 Solarity of the Set of Generalized Rational Fractions

7.11 Approximative Properties of Sets Lying in a Subspace

7.12 Approximation by Products

8 Existence of Chebyshev Subspaces

8.1 Chebyshev Subspaces in Finite-Dimensional Spaces

8.2 Chebyshev Subspaces in Infinite-Dimensional Spaces

8.3 Finite-Dimensional Chebyshev Subspaces in L1(µ).

9 Efimov-Stechkin Spaces. Uniform Convexity and Uniform Smoothness. Uniqueness and Strong Uniqueness of Best Approximation in Uniformly Convex Spaces

9.1 Efimov-Stechkin Spaces

9.2 Uniformly Convex Spaces

9.3 Uniqueness of Best Approximation by Convex Closed Sets …

9.4 Strong Uniqueness in Uniformly Convex Spaces

9.5 Uniformly Smooth Spaces

10 Solarity of Chebyshev Sets

10.1 Solarity of Boundedly Compact Chebyshev Sets

10.2 Relations Between Classes of Suns

10.3 Solarity of Chebyshev Sets

10.3.1 Solarity of Chebyshev Sets with Continuous Metric Projection

10.4 Solarity and Structural Properties of Sets

10.4.1 Solarity of Monotone Path-Connected Chebyshev Sets

10.4.2 Acyclicity and Cell-Likeness of Sets

10.4.3 Solarity of Boundedly Compact P-Acyclic Sets

11 Rational Approximation

11.1 Existence of a Best Rational Approximation

11.2 Characterization of Best Rational Approximation in the Space C[a,b]

11.3 Rational Lp-Approximation

11.4 Existence of Best Approximation by Generalized Rational Fractions

11.5 Characterization of Best Generalized Rational Approximation

11.6 Uniqueness of General Rational Approximation

11.7 Continuity of the Best Rational Approximation Operator

11.8 Notes on Algorithms of Rational Approximations

12 Haar Cones and Varisolvency

12.1 Properties of Haar Cones. Uniqueness …

12.2 Alternation Theorem for Haar Cones

12.3 Varisolvency

12.3.1 Uniqueness of Best Approximation by Varisolvent Sets

12.3.2 Regular and Singular Points in Approximation by Varisolvent Sets

13 Approximation of Vector-Valued Functions

13.1 Approximation of Abstract Functions. Interpolation and Uniqueness

13.2 Uniqueness of Best Approximation in the Mean for Vector-Valued Functions

13.3 On the Haar Condition for Systems of Vector-Valued Functions.

13.4 Approximation of Vector-Valued Functions by Polynomials

13.5 Some Applications of Vector-Valued Approximation

14 The Jung Constant

14.1 Definition of the Jung Constant

14.2 The Measure of Nonconvexity of a Space and the Jung Constant

14.3 The Jung Constant and Fixed Points of Condensing and Nonexpansive Maps

14.4 On an Approximate Solution of the Equation f(x)=x

14.5 On the Jung Constant of the Space ell1n

14.6 The Jung Constant and the Jackson Constant

14.7 The Relative Jung Constant

14.8 The Jung Constant of a Pair of Spaces

14.9 Some Remarks on Intersections of Convex Sets. Relation to the Jung Constant

15 Chebyshev Centre of a Set. The Problem of Simultaneous Approximation of a Class by a Singleton Set

15.1 Chebyshev Centre of a Set

15.2 Chebyshev Centres and Spans

15.3 Chebyshev Centre in the Space C(Q)

15.4 Existence of a Chebyshev Centre in Normed Spaces

15.4.1 Quasi-uniform Convexity and Existence of Chebyshev Centres

15.5 Uniqueness of a Chebyshev Centre

15.5.1 Uniqueness of a Chebyshev Centre of a Compact Set

15.5.2 Uniqueness of a Chebyshev Centre of a Bounded Set

15.6 Stability of the Chebyshev-Centre Map

15.6.1 Stability of the Chebyshev-Centre Map in Arbitrary Normed Spaces

15.6.2 Quasi-uniform Convexity and Stability of the Chebyshev-Centre Map

15.6.3 Stability of the Chebyshev-Centre Map in Finite-Dimensional Polyhedral Spaces

15.6.4 Stability of the Chebyshev-Centre Map in C(Q)-Spaces

15.6.5 Stability of the Chebyshev-Centre Map in Hilbert and Uniformly Convex Spaces

15.6.6 Stability of the Self-Chebyshev-Centre Map

15.6.7 Upper Semicontinuity of the Chebyshev-Centre Map and the Chebyshev-Near-Centre Map

15.6.8 Lipschitz Selection of the Chebyshev-Centre Map

15.6.9 Discontinuity of the Chebyshev-Centre Map.

15.7 Characterization of a Chebyshev Centre. Decomposition Theorem

15.8 Chebyshev Centres That Are Not Farthest Points

15.9 Smooth and Continuous Selections of the Chebyshev-Near-Centre Map

15.10 Algorithms and Applied Problems Connected with Chebyshev Centres

16 Width. Approximation by a Family of Sets

16.1 Problems in Recovery and Approximation Leading to Widths

16.2 Definitions of Widths

16.3 Fundamental Properties of Widths

16.4 Evaluation of Widths of ellp-Ellipsoids

16.5 Dranishnikov-Shchepin Widths and Their Relation to the CE-Problem

16.6 Bernstein Widths in the Spaces Linfty[0,1]

16.7 Widths of Function Classes

16.7.1 Definition of the Information Width

16.7.2 Estimates for Information Kolmogorov Widths

16.7.3 Some Exact Inequalities Between Widths. Projection Constants

16.7.4 Some Order Estimates and Duality of Information Width

16.7.5 Some Order Estimates for Information Kolmogorov Widths of Finite-Dimensional Balls

16.7.6 Order Estimates for Information Kolmogorov Widths of Function Classes

16.8 Relation Between the Jung Constant and Widths of Sets

16.9 Sequence of Best Approximations

17 Approximative Properties of Arbitrary Sets in Normed Linear Spaces. Almost Chebyshev Sets and Sets of Almost Uniqueness

17.1 Approximative Properties of Arbitrary Sets

17.2 Sets in Strictly Convex Spaces

17.3 Constructive Characteristics of Spaces

17.4 Sets in Locally Uniformly Convex Spaces

17.5 Sets in Uniformly Convex Spaces

17.6 Examples

17.7 Density and Category Properties of the Sets E(M), AC(M), and T(M)

17.8 Category Properties of the Set U(M)

17.9 Other Characteristics for the Size of Approximatively Defined Sets

17.10 The Farthest-Point Problem

17.11 Classes of Small Sets (Zk)

17.12 Contingent.

17.13 Zajíček-Smallness of the Classes of Sets R(M) and R*(M).

Notes:

Description based on print version record.

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Alimov, Alexey R. Geometric Approximation Theory

ISBN:

3-030-90951-4

OCLC:

1327748351