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)
17.14 Zajíček-Smallness of the Classes of Sets Rk(M) in Euclidean Spaces
17.15 Almost Chebyshev Sets
17.16 Almost Chebyshev Systems of Continuous Functions
A Chebyshev Systems of Functions in the Spaces C, Cn, and Lp
A.1 Statement of the Problem
A.2 Basic Facts About Structural Formulas for Extended Polynomials of Least Deviation from Zero
A.3 Further Results and Inequalities for Derivatives
A.4 Some Facts from Convex Analysis
A.5 Tests for Least Deviation from Zero for Extended Polynomials from Chebyshev Spaces
B Radon, Helly, and Carathéodory Theorems. Decomposition Theorem
B.1 Radon, Helly, and Carathéodory Theorems
B.2 Decomposition Theorem
C Some Open Problems
Index