An Introduction to the Approximation of Functions

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ISBN: 9780486640693 | 0486640698
Cover: Paperback
Copyright: 11/18/2010

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This graduate-level text offers a concise but wide-ranging introduction to methods of approximating continuous functions by functions depending only on a finite number of parameters. It places particular emphasis on approximation by polynomials and not only discusses the theoretical underpinnings of many common algorithms but also demonstrates their practical applications. 1969 edition.

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Introduction	p. 1
General Existence and Uniqueness of Approximations
Exercises	p. 8
Uniform Approximation	p. 11
Uniform Approximation by Polynomials	p. 11
The Weierstrass Theorem and Bernstein Polynomial Approximation	p. 11
Jackson's Theorems	p. 17
Characterization of Best Approximations	p. 24
Approximation on a Finite Set of Points	p. 33
Computational Methods	p. 40
The Exchange Method	p. 40
Linear Programming	p. 42
Exercises	p. 43
Least-Squares Approximation	p. 48
Approximation on an Interval	p. 48
The Jacobi Polynomials	p. 52
Approximation on a Finite Set of Points	p. 55
Effectiveness as a Uniform Approximation	p. 57
Exercises	p. 61
Least-First-Power Approximation	p. 66
Approximation on an Interval	p. 66
Approximation on a Finite Set of Points	p. 73
Some Computational Aspects	p. 78
Exercises	p. 83
Polynomial and Spline Interpolation	p. 87
General Results	p. 87
The Size of the Lebesgue Constants	p. 90
Interpolating Polynomials as Least-Squares and Least-First-Power Approximations	p. 101
Interpolation and Approximation by Splines	p. 104
Spline Interpolation	p. 104
Some Extremal Properties of Splines	p. 108
Uniform Approximation by Splines	p. 110
Least-Squares Approximation by Splines	p. 113
Exercises	p. 114
Approximation and Interpolation by Rational Functions	p. 120
Existence, Characterization, and Uniqueness	p. 120
Degree of Approximation	p. 128
Finite Point Sets	p. 130
Rational Interpolation	p. 132
Computing a Best Approximation	p. 135
Exercises	p. 138
Bibliography	p. 143
Index	p. 149
Table of Contents provided by Ingram. All Rights Reserved.

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An Introduction to the Approximation of Functions

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