GENERAL MATHEMATICS
CONTENTS CHAPTER 1 APPROXIMATION OF CONTINUOUS FUNCTIONS IN NORMED SPACES 1 1 INTRODUCTION 1 2 SOME REMARKS ON THE MEANING OF THE WORD “SIMPLE” CHOOSING THE APPROXIMATION 1 2 1 SPLINES 1 3 THE CHOICE OF THE NORM IN ORDER TO SPECIFY THE ERROR 1 4 OPTIMALITY WITH RESPECT TO A NORM 1 4 1 EXISTENCE OF AN OPTIMAL SOLUTION 1 4 2 UNIQUENESS OF THE OPTIMAL SOLUTION 1 4 3 EXAMPLES 1 5 CHARACTERIZING THE OPTIMAL SOLUTION 1 5 1 THE HILBERTIAN CASE 1 5 2 THE NON-HILBERTIAN CASE 1 5 3 OPTIMIZATION, LP NORMS AND ROBUSTNESS CHAPTER 2 CHEBYSHEV SYSTEMS 2 1 INTRODUCTION 2 2 FROM THE CLASSICAL POLYNOMIALS TO THE GENERALIZED ONES 2 2 1 EXAMPLES OF CHEBYSHEV SYSTEMS 2 3 PROPERTIES OF A CHEBYSHEV SYSTEM 2 3 1 VECTOR-TYPE PROPERTIES 2 3 2 CHEBYSHEV SYSTEMS AND INTERPOLATION 2 3 3 ROOTS OF THE GENERALIZED POLYNOMIALS CHAPTER 3 UNIFORM APPROXIMATIONS IN A NORMED SPACE 3 1 INTRODUCTION 3 2 CHARACTERIZATION OF THE BEST UNIFORM APPROXIMATION IN A NORMED SPACE 3 2 1 THE HAAR–KOLMOGOROV THEOREM 3 2 2 THE GENERALIZED BOREL – CHEBYSHEV THEOREM 3 2 3 OSCILLATION PROPERTIES OF THE BEST UNIFORM APPROXIMATION CHAPTER 4 CALCULATION OF THE BEST UNIFORM APPROXIMATION IN A CHEBYSHEV SYSTEM 4 1 SOME PRELIMINARY RESULTS 4 2 FUNCTIONAL CONTINUITY OF THE APPROXIMATION SCHEME 4 3 PROPERTY OF THE UNIFORM APPROXIMATION ON A ?NITE COLLECTION OF POINTS IN [A, B] 4 4 ALGORITHM OF DE LA VALLÉE POUSSIN 4 5 ALGORITHM OF REMEZ CHAPTER 5 OPTIMAL EXTRAPOLATION DESIGN FOR THE CHEBYSHEV REGRESSION 5 1 INTRODUCTION 5 2 THE MODEL AND GAUSS-MARKOV ESTIMATOR 5 2 1 DESCRIPTION OF THE DATASET 5 3 AN EXPRESSION OF THE EXTRAPOLATED VALUE THROUGH AN ORTHOGONALIZATION PROCEDURE 5 4 THE GAUSS-MARKOV ESTIMATOR OF THE EXTRAPOLATED VALUE 5 5 THE OPTIMAL EXTRAPOLATION DESIGN FOR THE CHEBYSHEV REGRESSION 5 5 1 THE SUPPORT OF THE OPTIMAL DESIGN 5 5 2 THE FREQUENCIES OF THE OPTIMAL DESIGN 5 5 3 IDENTI?CATION OF THE OPTIMAL DESIGN CHAPTER 6 OPTIMAL DESIGN FOR LINEAR FORMS OF THE PARAMETERS IN A CHEBYSHEV REGRESSION 6 1 OUTLOOK AND NOTATIONS 6 2 MATRIX OF MOMENTS 6 3 ESTIMABLE FORMS 6 4 MATRIX OF MOMENTS AND GAUSS-MARKOV ESTIMATORS OF A LINEAR FORM 6 4 1 MATRICES OF MOMENTS AND ESTIMABLE LINEAR FORMS 6 4 2 AN ALTERNATIVE FORM OF THE LOWER BOUND OF THE VARIANCE OF THE ESTIMATOR OF THE C FORM 6 5 GEOMETRIC INTERPRETATION OF ESTIMABILITY: ELFVING SET 6 5 1 ESTIMABLE FORMS AND A CONVEX SUBSET OF THE REGRESSION RANGE; THE ELFVING SET 6 5 2 GEOMETRY OF THE ELFVING SET 6 5 3 THE RELATION BETWEEN CYLINDERS AND THE VARIANCE OF THE ESTIMATOR OF THE C-FORM 6 5 4 LOWER BOUND FOR THE VARIANCE 6 5 5 THE LOWER BOUND CAN BE ACHIEVED 6 6 ELFVING THEOREM 6 7 AN INTUITIVE APPROACH TO ELFVING THEOREM 6 8 EXTENSION OF HOEL–LEVINE RESULT: OPTIMAL DESIGN FOR A LINEAR C-FORM CHAPTER 7 SPECIAL TOPICS AND EXTENSIONS 7 1 INTRODUCTION 7 2 THE GAUSS–MARKOV THEOREM IN VARIOUS CONTEXTS 7 2 1 THE GAUSS–MARKOV THEOREM FOR LINEAR TRANSFORMATIONS OF THE PARAMETER UNDER I I D ERRORS 7 2 2 GAUSS–MARKOV THEOREM FOR HETEROSCEDASTIC MODELS WITH CORRELATION 7 2 3 THE GAUSS–MARKOV THEOREM AND THE LOEWNER ORDER ON QUADRATIC FORMS 7 3 CRITERIONS FOR OPTIMAL DESIGNS 7 3 1 INTRODUCTION 7 3 2 SOME SPECI?C CRITERIONS 7 4 G–OPTIMAL INTERPOLATION AND EXTRAPOLATION DESIGNS FOR THE CHEBYSHEV REGRESSION 7 4 1 CRITERIA FOR OPTIMALITY 7 4 2 DESIGN WITH MINIMAL UNIFORM VARIANCE FOR A CHEBYSHEV REGRESSION 7 5 SOME QUESTIONS PERTAINING TO THE MODEL 7 5 1 LINEAR HETEROSCEDASTIC MODELS 7 5 2 NONLINEAR MODELS, ESTIMATORS AND OPTIMAL DESIGNS 7 6 HYPOTHESES PERTAINING TO THE REGRESSOR 7 6 1 REGRESSOR IN A LINEAR SPACE WITH UNKNOWN ?NITE DIMENSION 7 6 2 AN EXTENSION TO THE CASE OF ANALYTIC REGRESSORS 7 6 3 ON THE CHOICE OF THE DEGREE OF THE POLYNOMIAL MODEL 7 7 A FEW QUESTIONS PERTAINING TO THE SUPPORT OF THE OPTIMAL DESIGN FOR EXTRAPOLATION 7 7 1 PRELIMINARY RESULTS AND NOTATION 7 7 2 OPTIMAL DESIGNS WHOSE SUPPORT IS THE CHEBYSHEV SET OF POINTS 7 8 THE PROOFS OF SOME TECHNICAL RESULTS 7 8 1 PROOF OF PROPOSITION 7 1 7 8 2 PROOF OF THEOREM 7 17 CHAPTER 8 MULTIVARIATE MODELS AND ALGORITHMS 8 1 INTRODUCTION 8 2 MULTIVARIATE MODELS 8 2 1 NOTATION 8 2 2 PREDICTORS AND THEIR VARIANCE 8 2 3 SOME CONSTRUCTION OF MULTIVARIATE MODELS 8 3 OPTIMALITY CRITERIONS AND SOME OPTIMAL DESIGNS 8 3 1 CRITERIONS FOR OPTIMALITY AND CHARACTERIZATION OF THE OPTIMAL DESIGN 8 3 2 D?OPTIMALITY, DIRECT SUM AND TENSOR PRODUCT 8 4 ALGORITHMS 8 4 1 GENERAL ASPECTS 8 4 2 SPECI?C ALGORITHMS