Preface ........................................................ xi
Preface to the 3rd Edition ..................................... xv
Chapter 1 Introduction .......................................... 1
1.1 Approximation Problems ..................................... 1
1.2 Polynomials ................................................ 2
1.3 Piecewise Polynomials ...................................... 3
1.4 Spline Functions ........................................... 5
1.5 Function Classes and Computers ............................. 7
1.6 Historical Notes ........................................... 9
Chapter 2 Preliminaries ....................................... 12
2.1 Function Classes .......................................... 12
2.2 Taylor Expansions and the Green's Function ................ 14
2.3 Matrices and Determinants ................................. 19
2.4 Sign Changes and Zeros .................................... 24
2.5 Tchebycheff Systems ....................................... 29
2.6 Weak Tchebycheff Systems .................................. 36
2.7 Divided Differences ....................................... 45
2.8 Moduli of Smoothness ...................................... 54
2.9 The K-Functional .......................................... 59
2.10 n-Widths .................................................. 70
2.11 Periodic Functions ........................................ 75
2.12 Historical Notes .......................................... 76
2.13 Remarks ................................................... 79
Chapter 3 Polynomials ......................................... 83
3.1 Basic Properties .......................................... 83
3.2 Zeros and Determinants .................................... 85
3.3 Variation-Diminishing Properties .......................... 89
3.4 Approximation Power of Polynomials ........................ 91
3.5 Whitney-Type Theorems ..................................... 97
3.6 The Inflexibility of Polynomials ......................... 101
3.7 Historical Notes ......................................... 103
3.8 Remarks .................................................. 105
Chapter 4 Polynomial Splines ................................. 108
4.1 Basic Properties ......................................... 108
4.2 Construction of a Local Basis ............................ 112
4.3 B-Splines ................................................ 118
4.4 Equally Spaced Knots ..................................... 134
4.5 The Perfect B-Spline ..................................... 139
4.6 Dual Bases ............................................... 142
4.7 Zero Properties .......................................... 154
4.8 Matrices and Determinants ................................ 165
4.9 Variation-Diminishing Properties ......................... 177
4.10 Sign Properties of the Green's Function .................. 180
4.11 Historical Notes ......................................... 181
4.12 Remarks .................................................. 185
Chapter 5 Computational Methods .............................. 189
5.1 Storage and Evaluation ................................... 189
5.2 Derivatives .............................................. 195
5.3 The Piecewise Polynomial Representation .................. 197
5.4 Integrals ................................................ 199
5.5 Equally Spaced Knots ..................................... 204
5.6 Historical Notes ......................................... 207
5.7 Remarks .................................................. 208
Chapter 6 Approximation Power of Splines ..................... 210
6.1 Introduction ............................................. 210
6.2 Piecewise Constants ...................................... 212
6.3 Piecewise Linear Functions ............................... 220
6.4 Direct Theorems .......................................... 223
6.5 Direct Theorems in Intermediate Spaces ................... 233
6.6 Lower Bounds ............................................. 236
6.7 n-Widths ................................................. 239
6.8 Inverse Theory for p = ∞ ................................. 240
6.9 Inverse Theory for 1 ≤ p < ∞ ............................. 252
6.10 Historical Notes ......................................... 261
6.11 Remarks .................................................. 264
Chapter 7 Approximation Power of Splines (Free Knots) ........ 268
7.1 Introduction ............................................. 268
7.2 Piecewise Constants ...................................... 270
7.3 Variational Moduli of Smoothness ......................... 276
7.4 Direct and Inverse Theorems .............................. 278
7.5 Saturation ............................................... 283
7.6 Saturation Classes ....................................... 286
7.7 Historical Notes ......................................... 293
7.8 Remarks .................................................. 294
Chapter 8 Other Spaces of Polynomial Splines ................. 297
8.1 Periodic Splines ......................................... 297
8.2 Natural Splines .......................................... 309
8.3 g-Splines ................................................ 316
8.4 Monosplines .............................................. 330
8.5 Discrete Splines ......................................... 342
8.6 Historical Notes ......................................... 360
8.7 Remarks .................................................. 362
Chapter 9 Tchebycheffian Splines ............................. 363
9.1 Extended Complete Tchebycheff Systems .................... 363
9.2 A Green's Function ....................................... 373
9.3 Tchebycheffian Spline Functions .......................... 378
9.4 Tchebycheffian B-Splines ................................. 380
9.5 Zeros of Tchebycheffian Splines .......................... 388
9.6 Determinants and Sign Changes ............................ 390
9.7 Approximation Power of T-Splines ......................... 393
9.8 Other Spaces of Tchebycheffian Splines ................... 395
9.9 Exponential and Hyperbolic Splines ....................... 405
9.10 Canonical Complete Tchebycheff Systems ................... 407
9.11 Discrete Tchebycheffian Splines .......................... 411
9.12 Historical Notes ......................................... 417
Chapter 10 L-Splines .......................................... 420
10.1 Linear Differential Operators ............................ 420
10.2 A Green's Function ....................................... 424
10.3 L-Splines ................................................ 429
10.4 A Basis of Tchebycheffian B-Splines ...................... 433
10.5 Approximation Power of L-Splines ......................... 438
10.6 Lower Bounds ............................................. 440
10.7 Inverse Theorems and Saturation .......................... 444
10.8 Trigonometric Splines .................................... 452
10.9 Historical Notes ......................................... 459
10.10 Remarks ................................................. 461
Chapter 11 Generalized Splines ................................ 462
11.1 A General Space of Splines ............................... 462
11.2 A One-Sided Basis ........................................ 466
11.3 Constructing a Local Basis ............................... 470
11.4 Sign Changes and Weak Tchebycheff Systems ................ 472
11.5 A Nonlinear Space of Generalized Splines ................. 478
11.6 Rational Splines ......................................... 479
11.7 Complex and Analytic Splines ............................. 480
11.8 Historical Notes ......................................... 482
Chapter 12 Tensor-Product Splines ............................. 484
12.1 Tensor-Product Polynomial Splines ........................ 484
12.2 Tensor-Product B-Splines ................................. 486
12.3 Approximation Power of Tensor-Product Splines ............ 489
12.4 Inverse Theory for Piecewise Polynomials ................. 492
12.5 Inverse Theory for Splines ............................... 497
12.6 Historical Notes ......................................... 499
Chapter 13 Some Multidimensional Tools ........................ 501
13.1 Notation ................................................. 501
13.2 Sobolev Spaces ........................................... 503
13.3 Polynomials .............................................. 506
13.4 Taylor Theorems and the Approximation Power of
Polynomials .............................................. 507
13.5 Moduli of Smoothness ..................................... 516
13.6 The K-Functional ......................................... 520
13.7 Historical Notes ......................................... 521
13.8 Remarks .................................................. 523
Supplement .................................................... 524
References .................................................... 534
New References ................................................ 559
Index ......................................................... 577
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