导语

  

内容提要

  

    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
  13.7  Historical Notes
  13.8  Remarks
Supplement
References
New References
Index