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Michel Broniatowski, Professor of Statistics, Universitei Pierre et Marie Curie, Paris, France.

Preface ix

Introduction xi

Chapter 1 Approximation of Continuous Functions in Normed Spaces 1

1.1 Introduction 1

1.2 Some remarks on the meaning of the word “simple” Choosing the approximation  2

1.3 The choice of the norm in order to specify the error  8

1.4 Optimality with respect to a norm 12

1.5 Characterizing the optimal solution.18

Chapter 2 Chebyshev Systems 27

2.1 Introduction 27

2.2 From the classical polynomials to the generalized ones 28

2.3 Properties of a Chebyshev system 34

Chapter 3 Uniform Approximations in a Normed Space 45

3.1 Introduction 45

3.2 Characterization of the best uniform approximation in a normed space 46

Chapter 4 Calculation of the Best Uniform Approximation in a Chebyshev System 69

4.1 Some preliminary results   69

4.2 Functional continuity of the approximation scheme   71

4.3 Property of the uniform approximation on a finite collection of points in [a, b] 74

4.4 Algorithm of de la Vallée Poussin80

4.5 Algorithm of Remez    80

Chapter 5 Optimal Extrapolation Design for the Chebyshev Regression   85

5.1 Introduction 85

5.2 The model and Gauss-Markov estimator  87

5.3 An expression of the extrapolated value through an orthogonalization procedure 91

5.4 The Gauss-Markov estimator of the extrapolated value      93

5.5 The Optimal extrapolation design for the Chebyshev regression   97

Chapter 6 Optimal Design for Linear Forms of the Parameters in a Chebyshev Regression 107

6.1 Outlook and notations 107

6.2 Matrix of moments    113

6.3 Estimable forms  118

6.4 Matrix of moments and Gauss-Markov estimators of a linear form 119

6.5 Geometric interpretation of estimability: Elfving set        133

6.6 Elfving theorem   148

6.7 An intuitive approach to Elfving theorem  154

6.8 Extension of Hoel–Levine result: optimal design for a linear c-form 160

Chapter 7 Special Topics and Extensions  169

7.1 Introduction 169

7.2 The Gauss–Markov theorem in various contexts 170

7.3 Criterions for optimal designs   178

7.4 G–optimal interpolation and extrapolation designs for the Chebyshev regression  188

7.5 Some questions pertaining to the model 209

7.6 Hypotheses pertaining to the regressor 225

7.7 A few questions pertaining to the support of the optimal design for extrapolation    229

7.8 The proofs of some technical results   239

Chapter 8 Multivariate Models and Algorithms 249

8.1 Introduction  249

8.2 Multivariate models 250

8.3 Optimality criterions and some optimal designs 257

8.4 Algorithms 266

Bibliography 289

Index 295