I don't know if math books are allowed to be described as "page turners" but that's how I found this book. I read it cover to cover and came away knowing most everything I wanted to about radial basis functions and how to use them. I found the coverage of material to be well-motivated, complete, accurate, and very well written. If I were picky I would say that the author gets a bit carried away with measure theory formalism, particularly in the characterization of positive functions, but that's a minor complaint about an excellent book. I'd say it is a must-have for the library of anyone interested in theory and application of RBFs.
Author(s): Holger Wendland
Series: Cambridge Monographs on Applied and Computational Mathematics
Publisher: Cambridge University Press
Year: 2004
Language: English
Pages: 348
Tags: Математика;Вычислительная математика;
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Surface reconstruction......Page 13
1.2 Fluid–structure interaction in aeroelasticity......Page 16
1.3 Grid-free semi-Lagrangian advection......Page 18
1.4 Learning from splines......Page 19
1.5 Approximation and approximation orders......Page 25
1.6 Notation......Page 27
1.7 Notes and comments......Page 28
2.1 The Mairhuber–Curtis theorem......Page 30
2.2 Multivariate polynomials......Page 31
3.1 Definition and basic properties......Page 36
3.2 Norming sets......Page 38
3.3 Existence for regions with cone condition......Page 40
3.4 Notes and comments......Page 46
4.1 Definition and characterization......Page 47
4.2 Local polynomial reproduction by moving least squares......Page 52
4.3 Generalizations......Page 55
4.4 Notes and comments......Page 56
5.1 Bessel functions......Page 58
5.2 Fourier transform and approximation by convolution......Page 66
5.3 Measure theory......Page 72
6.1 Definition and basic properties......Page 76
6.2 Bochner’s characterization......Page 79
6.3 Radial functions......Page 90
6.5 Notes and comments......Page 96
7 Completely monotone functions......Page 97
7.1 Definition and first characterization......Page 98
7.2 The Bernstein–Hausdorff–Widder characterization......Page 100
7.3 Schoenberg’s characterization......Page 105
7.4 Notes and comments......Page 108
8.1 Definition and basic properties......Page 109
8.2 An analogue of Bochner’s characterization......Page 115
8.3 Examples of generalized Fourier transforms......Page 121
8.4 Radial conditionally positive definite functions......Page 125
8.5 Interpolation by conditionally positive definite functions......Page 128
8.6 Notes and comments......Page 129
9.1 General remarks......Page 131
9.2 Dimension walk......Page 132
9.3 Piecewise polynomial functions with local support......Page 135
9.4 Compactly supported functions of minimal degree......Page 139
9.5 Generalizations......Page 142
9.6 Notes and comments......Page 144
10.1 Reproducing-kernel Hilbert spaces......Page 145
10.2 Native spaces for positive definite kernels......Page 148
10.3 Native spaces for conditionally positive definite kernels......Page 153
10.4 Further characterizations of native spaces......Page 162
10.5 Special cases of native spaces......Page 168
10.6 An embedding theorem......Page 179
10.7 Restriction and extension......Page 180
10.8 Notes and comments......Page 182
11.1 Power function and first estimates......Page 184
11.2 Error estimates in terms of the fill distance......Page 189
11.3 Estimates for popular basis functions......Page 195
11.4 Spectral convergence for Gaussians and (inverse) multiquadrics......Page 200
11.5 Improved error estimates......Page 203
11.6 Sobolev bounds for functions with scattered zeros......Page 206
11.7 Notes and comments......Page 216
12 Stability......Page 218
12.1 Trade-off principle......Page 220
12.2 Lower bounds for Lambdamin......Page 221
12.3 Change of basis......Page 227
12.4 Notes and comments......Page 234
13.1 Minimal properties of radial basis functions......Page 235
13.2 Abstract optimal recovery......Page 238
13.3 Notes and comments......Page 241
14 Data structures......Page 242
14.1 The fixed-grid method......Page 243
14.2 kd-Trees......Page 249
14.3 bd-Trees......Page 255
14.4 Range trees......Page 258
14.5 Notes and comments......Page 263
15.1 Fast multipole methods......Page 265
15.2 Approximation of Lagrange functions......Page 277
15.3 Alternating projections......Page 282
15.4 Partition of unity......Page 287
15.5 Multilevel methods......Page 292
15.6 A greedy algorithm......Page 295
15.8 Notes and comments......Page 299
16.1 Optimal recovery in Hilbert spaces......Page 301
16.2 Hermite–Birkhoff interpolation......Page 304
16.3 Solving PDEs by collocation......Page 308
16.4 Notes and comments......Page 318
17.1 Spherical harmonics......Page 320
17.2 Positive definite functions on the sphere......Page 322
17.3 Error estimates......Page 326
17.4 Interpolation on compact manifolds......Page 328
17.5 Notes and comments......Page 333
References......Page 335
Index......Page 346
