**Graduate Studies in Mathematics**

Volume: 101;
2000;
359 pp;
Hardcover

MSC: Primary 41;

Print ISBN: 978-0-8218-4798-5

Product Code: GSM/101

List Price: $73.00

Individual Member Price: $58.40

**Electronic ISBN: 978-1-4704-1165-7
Product Code: GSM/101.E**

List Price: $73.00

Individual Member Price: $58.40

#### Supplemental Materials

# A Course in Approximation Theory

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*Ward Cheney; Will Light*

This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multi-variable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable.

Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thin-plate splines, box splines, ridge functions, and convolutions.

An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.

Originally published by Brooks Cole/Cengage Learning as ISBN: 978-0-534-36224-9.

#### Table of Contents

# Table of Contents

## A Course in Approximation Theory

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents xiii14 free
- Chapter 1 Introductory Discussion of Interpolation 118 free
- Chapter 2 Linear Interpolation Operators 1128
- Chapter 3 Optimization of the Lagrange Operator 1835
- Chapter 4 Multivariate Polynomials 2542
- Chapter 5 Moving the Nodes 3249
- Chapter 6 Projections 3956
- Chapter 7 Tensor-Product Interpolation 4663
- Chapter 8 The Boolean Algebra of Projections 5168
- Chapter 9 The Newton Paradigm for Interpolation 5774
- Chapter 10 The Lagrange Paradigm for Interpolation 6279
- Chapter 11 Interpolation by Translates of a Single Function 7188
- Chapter 12 Positive Definite Functions 7794
- Chapter 13 Strictly Positive Definite Functions 87104
- Chapter 14 Completely Monotone Functions 94111
- Chapter 15 The Schoenberg Interpolation Theorem 101118
- Chapter 16 The Micchelli Interpolation Theorem 109126
- Chapter 17 Positive Definite Functions on Spheres 119136
- Chapter 18 Approximation by Positive Definite Functions 131148
- Chapter 19 Approximate Reconstruction of Functions and Tomography 141158
- Chapter 20 Approximation by Convolution 148165
- Chapter 21 The Good Kernels 157174
- Chapter 22 Ridge Functions 165182
- Chapter 23 Ridge Function Approximation via Convolutions 177194
- Chapter 24 Density of Ridge Functions 184201
- Chapter 25 Artificial Neural Networks 189206
- Chapter 26 Chebyshev Centers 197214
- Chapter 27 Optimal Reconstruction of Functions 202219
- Chapter 28 Algorithmic Orthogonal Projections 210227
- Chapter 29 Cardinal B-Splines and the Sinc Function 215232
- Chapter 30 The Golomb-Weinberger Theory 223240
- Chapter 31 Hilbert Function Spaces and Reproducing Kernels 232249
- Chapter 32 Spherical Thin-Plate Splines 246263
- Chapter 33 Box Splines 260277
- Chapter 34 Wavelets, I 272289
- Chapter 35 Wavelets, II 285302
- Chapter 36 Quasi-Interpolation 312329
- Bibliography 327344
- Index 355372
- Index of Symbols 359376 free
- Back Cover Back Cover1377

#### Readership

Graduate students and research mathematicians interested in approximation theory and applications.

#### Reviews

Working through this book provides an opportunity to review and apply areas of mathematics learned elsewhere, as well as to learn entirely new topics.

-- MAA Reviews

The textbook, a clear and concise work written by world-renowned experts in the field of approximation theory, will probe useful not only as a reference for professional mathematicians but also as a text for graduate students.

-- Mathematical Reviews