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