Preface Preface to the 3rd Edition Chapter Ⅰ Introduction 1.1 Approximation Problems 1.2 Polynomials 1.3 Piecewise Polynomials 1.4 Spline Functions 1.5 Function Classes and Computers 1.6 Historical Notes Chapter 2 Preliminaries 2.1 Function Classes 2.2 Taylor Expansions and the Green's Function 2.3 Matrices and Determinants 2.4 Sign Changes and Zeros 2.5 Tchebycheff Systems 2.6 Weak Tchebycheff Systems 2.7 Divided Differences 2.8 Moduli of Smoothness 2.9 The K-Functional 2.10 n-Widths 2.11 Periodic Functions 2.12 Historical Notes 2.13 Remarks Chapter 3 Polynomials 3.1 Basic Properties 3.2 Zeros and Determinants 3.3 Variation-Diminishing Properties 3.4 Approximation Power of Polynomials 3.5 Whitney-Type Theorems 3.6 The Inflexibility of Polynomials 3.7 Historical Notes 3.8 Remarks Chapter 4 Polynomial Splines 4.1 Basic Properties 4.2 Construction of a Local Basis 4.3 B-Splines 4.4 Equally Spaced Knots 4.5 The Perfect B-Spline 4.6 Dual Bases 4.7 Zero Properties 4.8 Matrices and Determinants 4.9 Variation-Diminishing Properties 4.10 Sign Properties of the Green's Function 4.11 Historical Notes 4.12 Remarks Chapter 5 Computational Methods 5.1 Storage and Evaluation 5.2 Derivatives 5.3 The Piecewise Polynomial Representation 5.4 Integrals
5.5 Equally Spaced Knots 5.6 Historical Notes 5.7 Remarks Chapter 6 Approximation Power of Splines 6.1 Introduction 6.2 Piecewise Constants 6.3 Piecewise Linear Functions 6.4 Direct Theorems 6.5 Direct Theorems in Intermediate Spaces 6.6 Lower Bounds 6.7 n-Widths 6.8 Inverse Theory for p=∞ 6.9 Inverse Theory for 1?p＜∞ 6.10 Historical Notes 6.11 Remarks Chapter 7 Approximation Power of Splines (Free Knots) 7.1 Introduction 7.2 Piecewise Constants 7.3 Variational Moduli of Smoothness 7.4 Direct and Inverse Theorems 7.5 Saturation 7.6 Saturation Classes 7.7 Historical Notes 7.8 Remarks Chapter 8 Other Spaces of Polynomial Spllnes 8.1 Periodic Splines 8.2 Natural Splines 8.3 g-Splines 8.4 Monosplines 8.5 Discrete Splines 8.6 Historical Notes 8.7 Remarks Chapter 9 Tchebycheffian Splines 9.1 Extended Complete Tchebycheff Systems 9.2 A Green's Function 9.3 Tchebycheffian Spline Functions 9.4 Tchebycheffian B-Splines 9.5 Zeros of Tchebycheffian Splines 9.6 Determinants and Sign Changes 9.7 Approximation Power of T-Splines 9.8 Other Spaces of Tchebycheffian Splines 9.9 Exponential and Hyperbolic Splines 9.10 Canonical Complete Tchebycheff Systems 9.11 Discrete Tchebycheffian Splines 9.12 Historical Notes Chapter 10 L-Splines 10.1 Linear Differential Operators 10.2 A Green's Function 10.3 L-Splines 10.4 A Basis of Tchebycheffian B-Splines
10.5 Approximation Power of L-Splines 10.6 Lower Bounds 10.7 Inverse Theorems and Saturation 10.8 Trigonometric Splines 10.9 Historical Notes 10.10 Remarks Chapter 11 Generalized Splines 11.1 A General Space of Splines 11.2 A One-Sided Basis 11.3 Constructing a Local Basis 11.4 Sign Changes and Weak Tchebycheff Systems 11.5 A Nonlinear Space of Generalized Splines 11.6 Rational Splines 11.7 Complex and Analytic Splines 11.8 Historical Notes Chapter 12 Tensor-Product Splines 12.1 Tensor-Product Polynomial Splines 12.2 Tensor-Product B-Splines 12.3 Approximation Power of Tensor-Product Splines 12.4 Inverse Theory for Piecewise Polynomials 12.5 Inverse Theory for Splines 12.6 Historical Notes Chapter 13 Some Multidimensional Tools 13.1 Notation 13.2 Sobolev Spaces 13.3 Polynomials 13.4 Taylor Theorems and the Approximation Power of Polynomials 13.5 Moduli of Smoothness 13.6 The K-Functional 13.7 Historical Notes 13.8 Remarks Supplement References New References Index
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