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 Vallee 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
