逼近论教程（英文版）

这部著作是研究和学习逼近论的优秀教材，适合研究生层次的教学需求。作者是此领域里的知名学者，在世界范围内享有较高声誉。本书的重点是关于多元逼近论的研究。书中的大部分内容都是第一次系统地出现在教材里，有许多仍是当今研究的题材。主要内容包括正定函数、径向基插值、薄板样条、神经网络、岭函数以及小波分析等。每一章都包含了现今科学、工程学、地球物理学、商业和经济等研究领域内的实际问题。

Ward Cheney于堪萨斯州立大学获得博士学位，现为得州大学奥斯汀分校数学系教授。他的研究方向主要是逼近论、数值分析和极值问题等。现已发表了大量的学术论文，他所著(或与人合著)的书有：《Introduction to Approximation Theory》(1966)、《Approximation Theory in Tensor Product Spaces》(1985)，《数值分析》(Numerical Analysis：Mathematics of Scientific Computing)(本书影印版已由机械工业出版社出版)等。

Preface

This book offers a graduate-level exposition of selected topics in modern approximation theory. A large portion of the book focuses on multivariate approximation theory,where much recent research is concentrated. Although our own interests have influenced the choice of topics, the text cuts a wide swath through modem approximation theory, as can be seen from the table of contents. We believe the book will be found suitable as a text for courses, seminars, and even solo study. Although the book is at the graduate level, it does not presuppose that the reader already has taken a course in approximation theory.

Topics of This Book

A central theme of the book is the problem of interpolating data by smooth multivafiable

functions. Several chapters investigate interesting families of functions that can be employed in this task; among them are the polynomials, the positive definite functions,and the radial basis functions. Whether these same families can be used, in general, for approximating functions to arbitrary precision is a natural question that follows; it is addressed in further chapters.

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. Here there are questions of limiting behavior of sequences of operators, just as there are questions about interpolating on larger and larger sets of nodes.

A major departure from our theme of multivariate approximation is found in the two chapters on univariate wavelets, which comprise a significant fraction of the book.

In our opinion wavelet theory is so important a development in recent times--and is so mathematically appealing--that we had to devote some space to expounding its basic principles.

The Style of This Book In style, we have tried to make the exposition as simple and clear as possible, electing to furnish proofs that are complete and relatively easy to read without the reader needing to resort to pencil and paper. Any reader who finds this style too prolix can proceed quickly over arguments and calculations that are routine. To paraphrase Shaw: We have done our best to avoid conciseness! We have also made considerable efforts to find simple ways to introduce and explain each topic. We hope that in doing so, we encourage readers to delve deeper into some areas. It should be borne in mind that further exploration of some topics may require more mathematical sophistication than is demanded by our treatment.

Organization of the Book

A word about the general plan of the book: we start with relatively elementary matters in a series of about ten short chapters that do not, in general, i-equire more of the reader than undergraduate mathematics (in the American university system). From that point on, the gradient gradually increases and the text becomes more demanding, although still largely self-contained. Perhaps the most significant demands made on the technical knowledge of the reader fall in the areas of measure theory and the Fourier transform.

We have freely made use of the Lebesgue function spaces, which bring into play such measure-theoretic results as the Fubini Theorem. Other results such as the Riesz Representation Theorem for bounded linear functionals on a space of continuous functions and the Plancherel Theorem for Fourier transforms also are employed without compunction; but we have been careful to indicate explicitly how these ideas come into play.Consequently, the reader can simply accept the claims about such matters as they arise.Since these theorems form a vital part of the equipment of any applied analyst, we are confident that readers will want to understand for themselves the essentials of these areas of mathematics. We recommend Rudin's Real and Complex Analysis (McGraw-

Hill, 1974) as a suitable source for acquiring the necessary measure theoretic ideas, and the book Functional Analysis (McGraw-Hill, 1973) by the same author as a good introduction to the circle of ideas connected with the Fourier transform.

Additional Reading

We call the reader's attention to the list of books on approximation theory that immediately precedes the main section of references in the bibliography. These books, in general, are concerned with what we may term the "classical" portion of approximation theory--understood to mean the parts of the subject that already were in place when the authors were students. As there are very few textbooks coveting recent theory, our book should help to fill that "much needed gap" as some wag phrased it years ago. This list of books emphasizes only the systematic textbooks for, the subject as a whole, not the specialized texts and monographs.

Acknowledgments

It is a pleasure to have this opportunity of thanking three agencies that supported out"research over the years when this book was being written: the Division of Scientific Affairs of NATO, the Deutsche Forschungsgemeinschaft, and the Science and Engineering Research Council of Great Britain. For their helpful reviews of our manuscript,we thank Robert Schaback, University of Gottingen, and Larry Schumaker, Vanderbilt University. We acknowledge also the contribution made by many students, who patiently listened to us expound the material contained in this book and who raised incisive questions. Students and colleagues in Austin, Leicester, Wurzburg, Singapore, and Canterbury (NZ) all deserve our thanks. Professor S. L. Lee was especially helpful.

A special word of thanks goes to Ms. Margaret Combs of the University of Texas Mathematics Department. She is a superb technical typesetter in the modem sense of the word, that is, an expert in TEX. She patiently created the TEX files for lecture notes,starting about six years ago, and kept up with the constant editing of these notes, which were to become the backbone of the book.

The staff of Brooks/Cole Publishing has been most helpful and professional in guiding this book to its publication. In particular, we thank Gary Ostedt, sponsoring editor; Ragu Raghavan, marketing representative; and Janet Hill, production editor, for their personal contact with us during this project.

How to Reach Us

Readers are encouraged to bring errors and suggestions to our attention. E-mail is excellentfor this purpose: our addresses are cheney @ math.utexas.edu and pwl @mcs.le.ac.uk. A website for the book is maintained at http:www, math.utexas.edu/user/cheney/ATBOOK.

Ward Cheney

Will Light

This book offers a graduate-level exposition of selected topics in modern approximation theory. A large portion of the book focuses on multivariate approximation theory,where much recent research is concentrated. Although our own interests have influenced the choice of topics, the text cuts a wide swath through modem approximation theory, as can be seen from the table of contents. We believe the book will be found suitable as a text for courses, seminars, and even solo study. Although the book is at the graduate level, it does not presuppose that the reader already has taken a course in approximation theory.

Topics of This Book

A central theme of the book is the problem of interpolating data by smooth multivafiable

functions. Several chapters investigate interesting families of functions that can be employed in this task; among them are the polynomials, the positive definite functions,and the radial basis functions. Whether these same families can be used, in general, for approximating functions to arbitrary precision is a natural question that follows; it is addressed in further chapters.

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. Here there are questions of limiting behavior of sequences of operators, just as there are questions about interpolating on larger and larger sets of nodes.

A major departure from our theme of multivariate approximation is found in the two chapters on univariate wavelets, which comprise a significant fraction of the book.

In our opinion wavelet theory is so important a development in recent times--and is so mathematically appealing--that we had to devote some space to expounding its basic principles.

The Style of This Book In style, we have tried to make the exposition as simple and clear as possible, electing to furnish proofs that are complete and relatively easy to read without the reader needing to resort to pencil and paper. Any reader who finds this style too prolix can proceed quickly over arguments and calculations that are routine. To paraphrase Shaw: We have done our best to avoid conciseness! We have also made considerable efforts to find simple ways to introduce and explain each topic. We hope that in doing so, we encourage readers to delve deeper into some areas. It should be borne in mind that further exploration of some topics may require more mathematical sophistication than is demanded by our treatment.

Organization of the Book

A word about the general plan of the book: we start with relatively elementary matters in a series of about ten short chapters that do not, in general, i-equire more of the reader than undergraduate mathematics (in the American university system). From that point on, the gradient gradually increases and the text becomes more demanding, although still largely self-contained. Perhaps the most significant demands made on the technical knowledge of the reader fall in the areas of measure theory and the Fourier transform.

We have freely made use of the Lebesgue function spaces, which bring into play such measure-theoretic results as the Fubini Theorem. Other results such as the Riesz Representation Theorem for bounded linear functionals on a space of continuous functions and the Plancherel Theorem for Fourier transforms also are employed without compunction; but we have been careful to indicate explicitly how these ideas come into play.Consequently, the reader can simply accept the claims about such matters as they arise.Since these theorems form a vital part of the equipment of any applied analyst, we are confident that readers will want to understand for themselves the essentials of these areas of mathematics. We recommend Rudin's Real and Complex Analysis (McGraw-

Hill, 1974) as a suitable source for acquiring the necessary measure theoretic ideas, and the book Functional Analysis (McGraw-Hill, 1973) by the same author as a good introduction to the circle of ideas connected with the Fourier transform.

Additional Reading

We call the reader's attention to the list of books on approximation theory that immediately precedes the main section of references in the bibliography. These books, in general, are concerned with what we may term the "classical" portion of approximation theory--understood to mean the parts of the subject that already were in place when the authors were students. As there are very few textbooks coveting recent theory, our book should help to fill that "much needed gap" as some wag phrased it years ago. This list of books emphasizes only the systematic textbooks for, the subject as a whole, not the specialized texts and monographs.

Acknowledgments

It is a pleasure to have this opportunity of thanking three agencies that supported out"research over the years when this book was being written: the Division of Scientific Affairs of NATO, the Deutsche Forschungsgemeinschaft, and the Science and Engineering Research Council of Great Britain. For their helpful reviews of our manuscript,we thank Robert Schaback, University of Gottingen, and Larry Schumaker, Vanderbilt University. We acknowledge also the contribution made by many students, who patiently listened to us expound the material contained in this book and who raised incisive questions. Students and colleagues in Austin, Leicester, Wurzburg, Singapore, and Canterbury (NZ) all deserve our thanks. Professor S. L. Lee was especially helpful.

A special word of thanks goes to Ms. Margaret Combs of the University of Texas Mathematics Department. She is a superb technical typesetter in the modem sense of the word, that is, an expert in TEX. She patiently created the TEX files for lecture notes,starting about six years ago, and kept up with the constant editing of these notes, which were to become the backbone of the book.

The staff of Brooks/Cole Publishing has been most helpful and professional in guiding this book to its publication. In particular, we thank Gary Ostedt, sponsoring editor; Ragu Raghavan, marketing representative; and Janet Hill, production editor, for their personal contact with us during this project.

How to Reach Us

Readers are encouraged to bring errors and suggestions to our attention. E-mail is excellentfor this purpose: our addresses are cheney @ math.utexas.edu and pwl @mcs.le.ac.uk. A website for the book is maintained at http:www, math.utexas.edu/user/cheney/ATBOOK.

Ward Cheney

Will Light

Chapter 1

Introductory Discussion of Interpolation 1

Chapter 2

Linear Interpolation Operators 11

Chapter 3

Optimization of the Lagrange Operator 18

Chapter 4

Multivariate Polynomials 25

Chapter 5

Moving the Nodes 32

Chapter 6

Projections 39

Chapter 7

Tensor-Product Interpolation 46

Chapter 8

The Boolean Algebra of Projections 51

Chapter 9

The Newton Paradigm for Interpolation 57

Chapter 10

The Lagrange Paradigm for Interpolation 62

Chapter 11

Interpolation by Translates of a Single Function 71

Chapter 12

Positive Definite Functions 77

Chapter 13

Strictly Positive Definite Functions 87

Chapter 14

Completely Monotone Functions 94

Chapter 15

The Schoenberg Interpolation Theorem 101

Chapter 16

The Micchelli Interpolation,Theorem 109

Chapter 17

Positive Definite Functions on Spheres 119

Chapter 18

Approximation by Positive Definite Functions 131

Chapter 19

Approximate Reconstruction of Functions and Tomography 141

Chapter 20

Approximation by Convolution 148

Chapter 21

The Good Kernels 157

Chapter 22

Ridge Functions 165

Chapter 23

Ridge Function Approximation via Convolutions 177

Chapter 24

Density of Ridge Functions 184

Chapter 25

Artificial Neural Networks 189

Chapter 26

Chebyshev Centers 197

Chapter 27

Optimal Reconstruction of Functions 202

Chapter 28

Algorithmic Orthogonal Projections 210

Chapter 29

Cardinal B-Splines and the Sinc Function 215

Chapter 30

The Golomb-Weinberger Theory 223

Chapter 31

Hilbert Function Spaces and Reproducing Kernels 232

Chapter 32

Spherical Thin-Plate Splines 246

Chapter 33

Box Splines 260

Chapter 34

Wavelets, I 272

Chapter 35

Wavelets, II 285

Chapter 36

Quasi-Interpolation 312

Bibliography 327

Index 355

Index of Symbols 359

Introductory Discussion of Interpolation 1

Chapter 2

Linear Interpolation Operators 11

Chapter 3

Optimization of the Lagrange Operator 18

Chapter 4

Multivariate Polynomials 25

Chapter 5

Moving the Nodes 32

Chapter 6

Projections 39

Chapter 7

Tensor-Product Interpolation 46

Chapter 8

The Boolean Algebra of Projections 51

Chapter 9

The Newton Paradigm for Interpolation 57

Chapter 10

The Lagrange Paradigm for Interpolation 62

Chapter 11

Interpolation by Translates of a Single Function 71

Chapter 12

Positive Definite Functions 77

Chapter 13

Strictly Positive Definite Functions 87

Chapter 14

Completely Monotone Functions 94

Chapter 15

The Schoenberg Interpolation Theorem 101

Chapter 16

The Micchelli Interpolation,Theorem 109

Chapter 17

Positive Definite Functions on Spheres 119

Chapter 18

Approximation by Positive Definite Functions 131

Chapter 19

Approximate Reconstruction of Functions and Tomography 141

Chapter 20

Approximation by Convolution 148

Chapter 21

The Good Kernels 157

Chapter 22

Ridge Functions 165

Chapter 23

Ridge Function Approximation via Convolutions 177

Chapter 24

Density of Ridge Functions 184

Chapter 25

Artificial Neural Networks 189

Chapter 26

Chebyshev Centers 197

Chapter 27

Optimal Reconstruction of Functions 202

Chapter 28

Algorithmic Orthogonal Projections 210

Chapter 29

Cardinal B-Splines and the Sinc Function 215

Chapter 30

The Golomb-Weinberger Theory 223

Chapter 31

Hilbert Function Spaces and Reproducing Kernels 232

Chapter 32

Spherical Thin-Plate Splines 246

Chapter 33

Box Splines 260

Chapter 34

Wavelets, I 272

Chapter 35

Wavelets, II 285

Chapter 36

Quasi-Interpolation 312

Bibliography 327

Index 355

Index of Symbols 359

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