Scattered Data Approximation,9780521843355

Scattered Data Approximation

by
ISBN13: 9780521843355
ISBN10: 0521843359
Format: Hardcover
Pub. Date: 2004-12-13
Publisher(s): Cambridge University Press
List Price: $184.75

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Summary

This complete self-contained introduction to the theory of scattered data approximation brings together much of the necessary background material in one source. It also provides students with complete proofs to the theory developed within.

Table of Contents

Preface ix
Applications and motivations
1(17)
Surface reconstruction
1(3)
Fluid-structure interaction in aeroelasticity
4(2)
Grid-free semi-Lagrangian advection
6(1)
Learning from splines
7(6)
Approximation and approximation orders
13(2)
Notation
15(1)
Notes and comments
16(2)
Haar spaces and multivariate polynomials
18(6)
The Mairhuber-Curtis theorem
18(1)
Multivariate polynomials
19(5)
Local polynomial reproduction
24(11)
Definition and basic properties
24(2)
Norming sets
26(2)
Existence for regions with cone condition
28(6)
Notes and comments
34(1)
Moving least squares
35(11)
Definition and characterization
35(5)
Local polynomial reproduction by moving least squares
40(3)
Generalizations
43(1)
Notes and comments
44(2)
Auxiliary tools from analysis and measure theory
46(18)
Bessel functions
46(8)
Fourier transform and approximation by convolution
54(6)
Measure theory
60(4)
Positive definite functions
64(21)
Definition and basic properties
64(3)
Bochner's characterization
67(11)
Radial functions
78(4)
Functions, kernels, and other norms
82(2)
Notes and comments
84(1)
Completely monotone functions
85(12)
Definition and first characterization
86(2)
The Bernstein-Hausdorff-Widder characterization
88(5)
Schoenberg's characterization
93(3)
Notes and comments
96(1)
Conditionally positive definite functions
97(22)
Definition and basic properties
97(6)
An analogue of Bochner's characterization
103(6)
Examples of generalized Fourier transforms
109(4)
Radial conditionally positive definite functions
113(3)
Interpolation by conditionally positive definite functions
116(1)
Notes and comments
117(2)
Compactly supported functions
119(14)
General remarks
119(1)
Dimension walk
120(3)
Piecewise polynomial functions with local support
123(4)
Compactly supported functions of minimal degree
127(3)
Generalizations
130(2)
Notes and comments
132(1)
Native spaces
133(39)
Reproducing-kernel Hilbert spaces
133(3)
Native spaces for positive definite kernels
136(5)
Native spaces for conditionally positive definite kernels
141(9)
Further characterizations of native spaces
150(6)
Special cases of native spaces
156(11)
An embedding theorem
167(1)
Restriction and extension
168(2)
Notes and comments
170(2)
Error estimates for radial basis function interpolation
172(34)
Power function and first estimates
172(5)
Error estimates in terms of the fill distance
177(6)
Estimates for popular basis functions
183(5)
Spectral convergence for Gaussians and (inverse) multiquadics
188(3)
Improved error estimates
191(3)
Sobolev bounds for functions with scattered zeros
194(10)
Notes and comments
204(2)
Stability
206(17)
Trade-off principle
208(1)
Lower bounds for λmin
209(6)
Change of basis
215(7)
Notes and comments
222(1)
Optimal recovery
223(7)
Minimal properties of radial basis functions
223(3)
Abstract optimal recovery
226(3)
Notes and comments
229(1)
Data structures
230(23)
The fixed-grid method
231(6)
kd-Trees
237(6)
bd-Trees
243(3)
Range trees
246(5)
Notes and comments
251(2)
Numerical methods
253(36)
Fast multipole methods
253(12)
Approximation of Lagrange functions
265(5)
Alternating projections
270(5)
Partition of unity
275(5)
Multilevel methods
280(3)
A greedy algorithm
283(4)
Concluding remarks
287(1)
Notes and comments
287(2)
Generalized interpolation
289(19)
Optimal recovery in Hilbert spaces
289(3)
Hermite-Birkhoff interpolation
292(4)
Solving PDEs by collocation
296(10)
Notes and comments
306(2)
Interpolation on spheres and other manifolds
308(15)
Spherical harmonics
308(2)
Positive definite functions on the sphere
310(4)
Error estimates
314(2)
Interpolation on compact manifolds
316(5)
Notes and comments
321(2)
References 323(11)
Index 334

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