Spline Functions: Basic Theory (Cambridge Mathematical Library)

✍ Scribed by Larry Schumaker

• Publisher: Cambridge University Press
• Year: 2007
• Tongue: English
• Leaves: 600
• Edition: 3
• Category: Library

✦ Synopsis

This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.

✦ Table of Contents

COVER
HALF-TITLE
TITLE
COPYRIGHT
DEDICATION
CONTENTS
PREFACE
PREFACE TO THE 3RD EDITION
1 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
Section 1.1
Section 1.2
Section 1.3
Section 1.4
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
Section 2.1
Section 2.2
Section 2.3
Section 2.4
Section 2.5
Section 2.6
Section 2.7
Section 2.8
Section 2.9
Section 2.10
2.13. REMARKS
Remark 2.1
Remark 2.2
Remark 2.3
Remark 2.4
Remark 2.5
Remark 2.6
Remark 2.8
Remark 2.9
3 POLYNOMIALS
3.1. BASIC PROPERTIES
3.2. ZEROS AND DETERMINANTS
3.3. VARIATION DIMINISIDNG PROPERTIES
3.4. APPROXIMATION POWER OF POLYNOMIALS
3.5. WHITNEY-TYPE THEOREMS
3.6. THE INFLEXIBILITY OF POLYNOMIALS
3.7. HISTORICAL NOTES
Section 3.1
Section 3.2
Section 3.3
Section 3.4
Section 3.5
Section 3.6
3.8. REMARKS
Remark 3.1
Remark 3.2
Remark 3.3
Remark 3.4
Remark 3.5
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 TIlE GREEN'S FUNCTION
4.11. HISTORICAL NOTES
Section 4.1
Section 4.2
Section 4.3
Section 4.4
Section 4.5
Section 4.6
Section 4.7
Section 4.8
Section 4.9
Section 4.10
4.12. REMARKS
Remark 4.1
Remark 4.2
Remark 4.3
Remark 4.4
Remark 4.5
Remark 4.6
Remark 4.7
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
Section 5.1
Section 5.2
Section 5.3
Section 5.4
5.7. REMARKS
Remark 5.1
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…
6.9. INVERSE THEORY FOR 1 < p…
6.10. HISTORICAL NOTES
Section 6.2
Section 6.3
Section 6.4
Section 6.5
Section 6.6
Section 6.8
Section 6.9
6.11. REMARKS
Remark 6.1
Remark 6.2
Remark 6.3
Remark 6.4
Remark 6.5
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
Section 7.1
Section 7.2
Section 7.3
Section 7.4
Section 7.5
Section 7.6
7.8. REMARKS
Remark 7.1
Remark 7.2
Remark 7.3
Remark 7.4
Remark 7.5
Remark 7.6
8 OTHER SPACES OF POLYNOMIAL SPLINES
8.1. PERIODIC SPLINES
8.2. NATURAL SPLINES
8.3. g-SPLINES
8.4. MONOSPLINES
8.5. DISCRETE SPLINES
8.6. HISTORICAL NOTES
Section 8.1
Section 8.2
Section 8.3
Section 8.4
Section 8.5
8.7 REMARKS
Remark 8.1
Remark 8.2
Remark 8.3
9 TCHEBYCHEFFKAN SPLKNES
9.1. EXTENDED COMPLETE TCHEBYCHEFF SYSTEMS
9.2. A GREEN'S FUNCTlON
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 TCHEBYCHEFFIAN 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
Section 9.1
Section 9.2
Section 9.3
Section 9.4
Section 9.5
Section 9.6
Section 9.7
Section 9.8
Section 9.9
Section 9.10
Section 9.11
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
Section 10.1
Section 10.2
Section 10.3
Section 10.4
Section 10.5
Section 10.6
Section 10.7
Section 10.8
10.10 REMARKS
Remark 10.1
Remark 10.2
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
Section 11.1
Section 11.2
Section 11.3
Section 11.4
Section 11.5
Section 11.7
12 TENSOR-PRODUCT SPLINES
12.1. TENSOR-PRODUCT POLYNOMIAL SPLINES
12.2. TENSOR-PRODUT B-SPLINES
12.3. APPROXIMATION POWER OF TENSOR-PRODUT SPLINES
12.4. INVERSE THEORY FOR PIECEWISE POLYNOMIALS
12.5. INVERSE THEORY FOR SPLINES
12.6. HISTORICAL NOTES
Section 12.1
Section 12.3
Section 12.4
Section 12.5
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
Section 13.1
Section 13.2
Section 13.3
Section 13.4
Section 13.5
Section 13.6
13.8. REMARKS
Remark 13.1
Remark 13.2
SUPPLEMENT
CHAPTER 2. PRELIMINARIES
CHAPTER 3. POLYNOMIALS
CHAPTER 4. POLYNOMIAL SPLINES
CHAPTER 6. APPROXIMATION POWER OF SPLINES
CHAPTER 8. OTHER SPACES OF POLYNOMIAL SPLINES
CHAPTER 9. TCHEBYCHEFFIAN SPLINES
CHAPTER 10. L-SPLINES
CHAPTER 11. GENERALIZED SPLINES
Other New Results
Recent Spline Books
REFERENCES
NEW REFERENCES
INDEX