Approximation of functions

Abstract

Possibility of Approximation: 1. Basic notions 2. Linear operators 3. Approximation theorems 4. The theorem of Stone 5. Notes Polynomials of Best Approximation: 1. Existence of polynomials of best approximation 2. Characterization of polynomials of best approximation 3. Applications of convexity 4. Chebyshev systems 5. Uniqueness of polynomials of best approximation 6. Chebyshev's theorem 7. Chebyshev polynomials 8. Approximation of some complex functions 9. Notes Properties of Polynomials and Moduli of Continuity: 1. Interpolation 2. Inequalities of Bernstein 3. The inequality of Markov 4. Growth of polynomials in the complex plane 5. Moduli of continuity 6. Moduli of smoothness 7. Classes of functions 8. Notes The Degree of Approximation by Trigonometric Polynomials: 1. Generalities 2. The theorem of Jackson 3. The degree of approximation of differentiable functions 4. Inverse theorems 5. Differentiable functions 6. Notes The Degree of Approximation by Algebraic Polynomials: 1. Preliminaries 2. The approximation theorems 3. Inequalities for the derivatives of polynomials 4. Inverse theorems 5. Approximation of analytic functions 6. Notes Approximation by Rational Functions. Functions of Several Variables: 1. Degree of rational approximation 2. Inverse theorems 3. Periodic functions of several variables 4. Approximation by algebraic polynomials 5. Notes Approximation by Linear Polynomial Operators: 1. Sums of de la Vallee-Poussin. Positive operators 2. The principle of uniform boundedness 3. Operators that preserve trigonometric polynomials 4. Trigonometric saturation classes 5. The saturation class of the Bernstein polynomials 6. Notes Approximation of Classes of Functions: 1. Introduction 2. Approximation in the space 3. The degree of approximation of the classes 4. Distance matrices 5. Approximation of the classes 6. Arbitrary moduli of continuity Approximation by operators 7. Analytic functions 8. Notes Widths: 1. Definitions and basic properties 2. Sets of continuous and differentiable functions 3. Widths of balls 4. Applications of theorem 2 5. Differential operators 6. Widths of the sets 7. Notes Entropy: 1. Entropy and capacity 2. Sets of continuous and differentiable functions 3. Entropy of classes of analytic functions 4. More general sets of analytic functions 5. Relations between entropy and widths 6. Notes Representation of Functions of Several Variables by Functions of One Variable: 1. The Theorem of Kolmogorov 2. The fundamental lemma 3. The completion of the proof 4. Functions not representable by superpositions 5. Notes Bibliography Index.