Spline Functions: Basic Theory (Cambridge Mathematical Library)

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TITLE
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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