This is an easily accessible book on the approximation of functions - simple and without unnecessary details, but complete enough to include the main results of the theory. Except for a few sections, only functions of a real variable are treated. This work can be used as a textbook for graduate or advanced undergraduate courses or for self-study. Included in the volume are Notes at the end of each chapter, Problems, and a selected Bibliography.
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.