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.
Author(s): Larry Schumaker
Series: Cambridge Mathematical Library
Edition: 3
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 600
COVER......Page 1
HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
DEDICATION......Page 7
CONTENTS......Page 9
PREFACE......Page 13
PREFACE TO THE 3RD EDITION......Page 17
1.1. APPROXIMATION PROBLEMS......Page 19
1.2. POLYNOMIALS......Page 20
1.3. PIECEWISE POLYNOMIALS......Page 21
1.4. SPLINE FUNCTIONS......Page 23
1.5. FUNCTION CLASSES AND COMPUTERS......Page 25
Section 1.3......Page 27
Section 1.4......Page 28
2.1. FUNCTION CLASSES......Page 30
2.2. TAYLOR EXPANSIONS AND THE GREEN'S FUNCTION......Page 32
2.3. MATRICES AND DETERMINANTS......Page 37
2.4. SIGN CHANGES AND ZEROS......Page 42
2.5. TCHEBYCHEFF SYSTEMS......Page 47
2.6. WEAK TCHEBYCHEFF SYSTEMS......Page 54
2.7. DIVIDED DIFFERENCES......Page 63
2.8. MODULI OF SMOOTHNESS......Page 72
2.9. THE K-FUNCTIONAL......Page 77
2.10. n-WIDTHS......Page 88
2.11. PERIODIC FUNCTIONS......Page 93
Section 2.2......Page 94
Section 2.6......Page 95
Section 2.9......Page 96
Remark 2.2......Page 97
Remark 2.5......Page 98
Remark 2.6......Page 99
Remark 2.9......Page 100
3.1. BASIC PROPERTIES......Page 101
3.2. ZEROS AND DETERMINANTS......Page 103
3.3. VARIATION DIMINISIDNG PROPERTIES......Page 107
3.4. APPROXIMATION POWER OF POLYNOMIALS......Page 109
3.5. WHITNEY-TYPE THEOREMS......Page 115
3.6. THE INFLEXIBILITY OF POLYNOMIALS......Page 119
Section 3.1......Page 121
Section 3.5......Page 122
Remark 3.2......Page 123
Remark 3.4......Page 124
Remark 3.5......Page 125
4.1. BASIC PROPERTIES......Page 126
4.2. CONSTRUCTION OF A LOCAL BASIS......Page 130
4.3. B-SPLINES......Page 136
4.4. EQUALLY SPACED KNOTS......Page 152
4.5. THE PERFECT B-SPLINE......Page 157
4.6. DUAL BASES......Page 160
4.7 ZERO PROPERTIES......Page 172
4.8. MATRICES AND DETERMINANTS......Page 183
4.9. VARIATION-DIMINISHING PROPERTIES......Page 195
4.10. SIGN PROPERTIES OF TIlE GREEN'S FUNCTION......Page 198
Section 4.2......Page 199
Section 4.3......Page 200
Section 4.6......Page 201
Section 4.8......Page 202
Remark 4.1......Page 203
Remark 4.3......Page 204
Remark 4.5......Page 205
Remark 4.7......Page 206
5.1. STORAGE AND EVALUATION......Page 207
5.2. DERIVATIVES......Page 213
5.3. THE PIECEWISE POLYNOMIAL REPRESENTATION......Page 215
5.4. INTEGRALS......Page 217
5.5. EQUALLY SPACED KNOTS......Page 222
Section 5.2......Page 225
Remark 5.1......Page 226
6.1. INTRODUCTION......Page 228
6.2. PIECEWISE CONSTANTS......Page 230
6.3. PIECEWISE LINEAR FUNCTIONS......Page 238
6.4. DIRECT THEOREMS......Page 241
6.5. DIRECT THEOREMS IN INTERMEDIATE SPACES......Page 251
6.6. LOWER BOUNDS......Page 254
6.7. n-WIDTHS......Page 257
6.8. INVERSE THEORY FOR…......Page 258
6.9. INVERSE THEORY FOR 1 < p…......Page 270
Section 6.4......Page 279
Section 6.5......Page 280
Section 6.8......Page 281
Remark 6.1......Page 282
Remark 6.3......Page 283
Remark 6.4......Page 284
Remark 6.5......Page 285
7.1. INTRODUCTION......Page 286
7.2. PIECEWISE CONSTANTS......Page 288
7.3 VARIATIONAL MODULI OF SMOOTHNESS......Page 294
7.4. DIRECT AND INVERSE THEOREMS......Page 296
7.5. SATURATION......Page 301
7.6. SATURATION CLASSES......Page 304
Section 7.3......Page 311
Remark 7.1......Page 312
Remark 7.3......Page 313
Remark 7.6......Page 314
8.1. PERIODIC SPLINES......Page 315
8.2. NATURAL SPLINES......Page 327
8.3. g-SPLINES......Page 334
8.4. MONOSPLINES......Page 348
8.5. DISCRETE SPLINES......Page 360
Section 8.2......Page 378
Section 8.4......Page 379
Remark 8.3......Page 380
9.1. EXTENDED COMPLETE TCHEBYCHEFF SYSTEMS......Page 381
9.2. A GREEN'S FUNCTlON......Page 391
9.3. TCHEBYCHEFFIAN SPLINE FUNCTIONS......Page 396
9.4. TCHEBYCHEFFIAN B-SPLINES......Page 398
9.5. ZEROS OF TCHEBYCHEFFIAN SPLINES......Page 406
9.6. DETERMINANTS AND SIGN CHANGES......Page 408
9.7. APPROXIMATION POWER OF TCHEBYCHEFFIAN SPLINES......Page 411
9.8. OTHER SPACES OF TCHEBYCHEFFIAN SPLINES......Page 413
9.9. EXPONENTIAL AND HYPERBOLIC SPLINES......Page 423
9.10. CANONICAL COMPLETE TCHEBYCHEFF SYSTEMS......Page 425
9.11. DISCRETE TCHEBYCHEFFIAN SPLINES......Page 429
Section 9.4......Page 435
Section 9.8......Page 436
Section 9.11......Page 437
10.1. LINEAR DIFFERENTIAL OPERATORS......Page 438
10.2. A GREEN'S FUNCTION......Page 442
10.3. L-SPLINES......Page 447
10.4. A BASIS OF TCHEBYCHEFFIAN B-SPLINES......Page 451
10.5. APPROXIMATION POWER OF L-SPLINES......Page 456
10.6. LOWER BOUNDS......Page 458
10.7. INVERSE THEOREMS AND SATURATION......Page 462
10.8. TRIGONOMETRIC SPLINES......Page 470
Section 10.3......Page 477
Section 10.8......Page 478
Remark 10.2......Page 479
11.1. A GENERAL SPACE OF SPLINES......Page 480
11.2. A ONE-SIDED BASIS......Page 484
11.3. CONSTRUCTING A LOCAL BASIS......Page 488
11.4. SIGN CHANGES AND WEAK TCHEBYCHEFF SYSTEMS......Page 490
11.5. A NONLINEAR SPACE OF GENERALIZED SPLINES......Page 496
11.6. RATIONAL SPLINES......Page 497
11.7. COMPLEX AND ANALYTIC SPLINES......Page 498
Section 11.2......Page 500
Section 11.7......Page 501
12.1. TENSOR-PRODUCT POLYNOMIAL SPLINES......Page 502
12.2. TENSOR-PRODUT B-SPLINES......Page 504
12.3. APPROXIMATION POWER OF TENSOR-PRODUT SPLINES......Page 507
12.4. INVERSE THEORY FOR PIECEWISE POLYNOMIALS......Page 510
12.5. INVERSE THEORY FOR SPLINES......Page 515
Section 12.3......Page 517
Section 12.5......Page 518
13.1. NOTATION......Page 519
13.2. SOBOLEV SPACES......Page 521
13.3. POLYNOMIALS......Page 524
13.4. TAYLOR THEOREMS AND THE APPROXIMATION POWER OF POLYNOMIALS......Page 525
13.5. MODULI OF SMOOTHNESS......Page 534
13.6. THE K-FUNCTIONAL......Page 538
Section 13.1......Page 539
Section 13.4......Page 540
Remark 13.2......Page 541
CHAPTER 3. POLYNOMIALS......Page 542
CHAPTER 4. POLYNOMIAL SPLINES......Page 543
CHAPTER 6. APPROXIMATION POWER OF SPLINES......Page 547
CHAPTER 9. TCHEBYCHEFFIAN SPLINES......Page 548
CHAPTER 10. L-SPLINES......Page 549
CHAPTER 11. GENERALIZED SPLINES......Page 550
Recent Spline Books......Page 551
REFERENCES......Page 552
NEW REFERENCES......Page 577
INDEX......Page 595
