It is necessary to estimate parameters by approximation and interpolation in many areas-from computer graphics to inverse methods to signal processing. Radial basis functions are modern, powerful tools which are being used more widely as the limitations of other methods become apparent. Martin Buhmann provides a complete analysis of radial basic functions from the theoretical and practical implementation viewpoints. He also includes a comprehensive bibliography.
Author(s): Martin D. Buhmann
Series: Cambridge Monographs on Applied and Computational Mathematics
Publisher: Cambridge University Press
Year: 2003
Language: English
Pages: 272
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
1 Introduction......Page 13
1.1 Radial basis functions......Page 14
1.2 Applications......Page 17
1.3 Contents of the book......Page 20
2.1 Invertibility of interpolation matrices......Page 23
2.2 Convergence analysis......Page 28
2.3.1 Central results about existence......Page 35
2.3.2 Properties of the Lagrange function......Page 38
2.4 Applications to PDEs......Page 41
3 General Methods for Approximation and Interpolation......Page 48
3.1 Polynomial schemes......Page 49
3.2 Piecewise polynomials......Page 53
3.3 General nonpolynomial methods......Page 57
4 Radial Basis Function Approximation on Infinite Grids......Page 60
4.1 Existence of interpolants......Page 61
4.2.1 Approximation orders on gridded data......Page 77
4.2.2 Quasi-interpolation versus Lagrange interpolation......Page 83
4.2.3 Upper bounds on approximation orders......Page 89
4.2.4 Approximation orders without Fourier transforms......Page 92
4.3 Numerical properties of the interpolation linear system......Page 101
4.4 Convergence with respect to parameters in the radial function......Page 107
5 Radial Basis Functions on Scattered Data......Page 111
5.1 Nonsingularity of interpolation matrices......Page 112
5.2.1 Variational theory of radial basis functions......Page 117
5.2.2 The reproducing kernel (semi-)Hilbert space......Page 120
5.2.3 Minimum norm interpolants......Page 127
5.2.4 The power functional and convergence estimates......Page 128
5.2.5 Further results......Page 143
5.3.1 General remarks......Page 148
5.3.2 Bounds on eigenvalues......Page 149
5.3.3 Lower bounds on matrix norms......Page 153
5.3.4 The uncertainty principle......Page 155
6.1 Introduction......Page 159
6.2 Wendland’s functions......Page 162
6.3 Another class of radial basis functions with compact support......Page 165
6.4 Convergence......Page 171
6.5 A unified class......Page 174
7 Implementations......Page 175
7.1 Introduction......Page 176
7.2 The BFGP algorithm and the new Krylov method......Page 179
7.2.1 Data structures and local Lagrange functions......Page 180
7.2.2 Description of the algorithm......Page 182
7.2.3 Convergence......Page 187
7.2.4 The Krylov subspace method......Page 189
7.3 The fast multipole algorithm......Page 195
7.4 Preconditioning techniques......Page 200
8.1 Introduction to least squares......Page 208
8.2 Approximation order results......Page 211
8.3 Discrete least squares......Page 213
8.4 Implementations......Page 219
8.5 Neural network applications......Page 220
9.1 Introduction to wavelets and prewavelets......Page 221
9.2 Basic definitions and constructions......Page 224
9.3.1 Definition of MRA and the prewavelets......Page 226
9.3.2 The fast wavelet transform FWT......Page 229
9.3.3 The refinement equation......Page 231
9.3.4 The Riesz basis property......Page 233
9.3.5 First constructions of prewavelets......Page 236
9.4.1 An example of a prewavelet construction......Page 238
9.4.2 Generalisations, review of further constructions......Page 239
10.1 Further results......Page 243
10.2 Open problems......Page 248
Appendix: Some Essentials on Fourier Transforms......Page 252
Commentary on the Bibliography......Page 255
Bibliography......Page 258
Index......Page 270
