Approximation of functions
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Bibliographic Information
Approximation of functions
AMS Chelsea Pub., 2005
2nd ed.
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Note
Bibliography: p. 179-184
Includes index
Reprint. Originally published: New York : Chelsea Pub. Co., c1966 , 1986 ; 2nd ed
Description and Table of Contents
Description
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.
Table of Contents
- 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.
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