本书是一部全面介绍单变量和张量积样条函数理论的经典著作，为便于读者理解，书中呈现了样条理论在诸多领域的应用，其中包括近似理论，计算机辅助几何设计，曲线和曲面设计与拟合，图像处理，微分方程的数值解，强调了该理论在商业和生物科学中的应用也日益广泛。本书主要面向应用分析、数值分析、计算科学和工程领域的研究生和科学工作者，也可作为样条理论、近似理论和数值分析等应用数学专业课教材或教学参考书。

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
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