Approximation of Functions

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American Mathematical Soc., 2005 - Mathematics - 188 pages
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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 Arnold 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.
 

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Contents

Possibility of Approximation
1
Polynomials of Best Approximation
16
Properties of Polynomials and Moduli of Continuity
36
The Degree of Approximation by Trigonometric Polynomials
54
The Degree of Approximation by Algebraic Polynomials
65
Approximation by Rational Functions Functions of Several Variables
81
Approximation by Linear Polynomial Operators
92
Approximation of Classes of Functions
111
Widths
132
Entropy
150
Representation of Functions of Several Variables by Functions of One Variable
168
Bibliography
179
Index
187
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