This is an easily accessible account of the approximation of functions. It is simple and without unnecessary details, but complete enough to include the classical results of the theory. With only a few exceptions, only functions of one real variable are considered. A major theme is the degree of uniform approximation by linear sets of functions. This encompasses approximations by trigonometric polynomials, algebraic polynomials, rational functions, and polynomial operators. The chapter on approximation by operators does not assume extensive knowledge of functional analysis. Two chapters cover the important topics of widths and entropy. The last chapter covers the solution by Kolmogorov and Arnol‴d of Hilbert's 13th problem. There are notes at the end of each chapter that give information about important topics not treated in the main text. Each chapter also has a short set of challenging problems, which serve as illustrations.
Front Cover Preface to the Second Edition Preface to the First Edition Contents Chapter 1. Possibility of Approximation 1. Basic Notions 2. Linear Operators 3. Approximation Theorems 4. The Theorem of Stone 5. Notes PROBLEMS Chapter 2. 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 PROBLEMS Chapter 3. 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 PROBLEMS Chapter 4. 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 PROBLEMS Chapter 5. 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 PROBLEMS Chapter 6. Approximation by Rational Functions. Functions of Several Variables 1. Degree of Rational Approximation 2. Further Theorems 3. Periodic Functions of Several Variables 4. Approximation by Algebraic Polynomials 5. Notes PROBLEMS Chapter 7. 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 PROBLEMS Chapter 8. Approximation of Classes of Functions 1. Introduction 2. Approximation in the Space L1 3. The Degree of Approximation of the Classes W*p 4. Distance Matrices 5. Approximation of the Classes ?? 6. Arbitrary Moduli of Continuity Approximation by Operators 7. Analytic Functions 8. Notes PROBLEMS Chapter 9. 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 Set R1 7. Notes PROBLEMS Chapter 10. Entropy 1. Entropy and Capacity 2. Sets of Continuous and Differentiable Functions 3. Entropy of Classes of Analytic Functions 4. General Sets of Analytic Functions 5. Relations between Entropy and Widths 6. Notes PROBLEMS Chapter 11. 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 PROBLEMS Bibliography Index Back Cover
