Course in Approximation Theory

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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.

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Editorial Reviews

The central theme of the graduate textbook is the problem of interpolating data by smooth multivariable functions. Several chapters investigate interesting families of functions that can be employed in this task. The book then moves on to the consideration of methods for concocting approximations, such as by convolutions, by neural nets, or by interpolation at more and more points. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780821847985
  • Publisher: American Mathematical Society
  • Publication date: 1/13/2009
  • Series: Graduate Studies in Mathematics Series, #101
  • Edition description: New Edition
  • Pages: 359
  • Product dimensions: 6.40 (w) x 9.30 (h) x 0.70 (d)

Meet the Author

U of Leicester, U.K.
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Table of Contents

Ch. 1 Introductory Discussion of Interpolation 1

Ch. 2 Linear Interpolation Operators 11

Ch. 3 Optimization of the Lagrange Operator 18

Ch. 4 Multivariate Polynomials 25

Ch. 5 Moving the Nodes 32

Ch. 6 Projections 39

Ch. 7 Tensor-Product Interpolation 46

Ch. 8 The Boolean Algebra of Projections 51

Ch. 9 The Newton Paradigm for Interpolation 57

Ch. 10 The Lagrange Paradigm for Interpolation 62

Ch. 11 Interpolation by Translates of a Single Function 71

Ch. 12 Positive Definite Functions 77

Ch. 13 Strictly Positive Definite Functions 87

Ch. 14 Completely Monotone Functions 94

Ch. 15 The Schoenberg Interpolation Theorem 101

Ch. 16 The Micchelli Interpolation Theorem 109

Ch. 17 Positive Definite Functions on Spheres 119

Ch. 18 Approximation by Positive Definite Functions 131

Ch. 19 Approximate Reconstruction of Functions and Tomography 141

Ch. 20 Approximation by Convolution 148

Ch. 21 The Good Kernels 157

Ch. 22 Ridge Functions 165

Ch. 23 Ridge Function Approximation via Convolutions 177

Ch. 24 Density of Ridge Functions 184

Ch. 25 Artificial Neural Networks 189

Ch. 26 Chebyshev Centers 197

Ch. 27 Optimal Reconstruction of Functions 202

Ch. 28 Algorithmic Orthogonal Projections 210

Ch. 29 Cardinal B-Splines and the Sinc Function 215

Ch. 30 The Golomb-Weinberger Theory 223

Ch. 31 Hilbert Function Spaces and Reproducing Kernels 232

Ch. 32 Spherical Thin-Plate Splines 246

Ch. 33 Box Splines 260

Ch. 34 Wavelets, I 272

Ch. 35 Wavelets, II 285

Ch. 36 Quasi-Interpolation 312

Bibliography 327

Index 355

Index of Symbols 359

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