A Course in Approximation Theory

by Cheney, E. Ward

• ISBN13: 9780534362249
• ISBN10: 0534362249
• Edition: 1st
• Format: Paperback
• Copyright: 1999-07-05
• Publisher: Brooks Cole

Summary

1. Introductory Discussion of Interpolation 2. Linear Interpolation Operators 3. Optimization of the Lagrange Operator 4. Multivariate Polynomials 5. Moving the Nodes 6. Projections 7. Tensor Product Interpolation 8. The Boolean Algebra of Projections 9. The Newton Paradigm for Interpolation 10. The Lagrange Paradigm for Interpolation 11. Interpolation by Translates of a Single Function 12. Positive Definite Functions 13. Strictly Positive Definite Functions 14. Completely Monotone Functions 15. The Schoenberg Interpolation Theorem 16. The Micchelli Interpolation Theorem 17. Positive Definite Functions of Spheres 18. Approximation by Positive Definite Functions 19. Approximate Reconstruction of Functions and Tomography 20. Approximation by Convolution 21. The Good Kernels 22. Ridge Functions 23. Ridge Function Approximation via Convolutions 24. Density of Ridge Functions 25. Artificial Neural Networks 26. Chebyshev Centers 27. Optimal Reconstruction of Functions 28. Algorithmic Orthogonal Projections 29. Cardinal B-Splines and the Sinc Function 30. The Golomb-Weinberger Theory 31. Hilbert Function Spaces, Reproducing Kernels 32. Spherical Splines 33. Box Splines 34. Wavelets, Part I 35. Wavelets, Part II 36. Quasi-Interpolation Bibliography / Index

Table of Contents

Introductory Discussion of Interpolation	1	(10)
Linear Interpolation Operators	11	(7)
Optimization of the Lagrange Operator	18	(7)
Multivariate Polynomials	25	(7)
Moving the Nodes	32	(7)
Projections	39	(7)
Tensor-Product Interpolation	46	(5)
The Boolean Algebra of Projections	51	(6)
The Newton Paradigm for Interpolation	57	(5)
The Lagrange Paradigm for Interpolation	62	(9)
Interpolation by Translates of a Single Function	71	(6)
Positive Definite Functions	77	(10)
Strictly Positive Definite Functions	87	(7)
Completely Monotone Functions	94	(7)
The Schoenberg Interpolation Theorem	101	(8)
The Micchelli Interpolation Theorem	109	(10)
Positive Definite Functions on Spheres	119	(12)
Approximation by Positive Definite Functions	131	(10)
Approximate Reconstruction of Functions and Tomography	141	(7)
Approximation by Convolution	148	(9)
The Good Kernels	157	(8)
Ridge Functions	165	(12)
Ridge Function Approximation via Convolutions	177	(7)
Density of Ridge Functions	184	(5)
Artificial Neural Networks	189	(8)
Chebyshev Centers	197	(5)
Optimal Reconstruction of Functions	202	(8)
Algorithmic Orthogonal Projections	210	(5)
Cardinal B-Splines and the Sinc Function	215	(8)
The Golomb-Weinberger Theory	223	(9)
Hilbert Function Spaces and Reproducing Kernels	232	(14)
Spherical Thin-Plate Splines	246	(14)
Box Splines	260	(12)
Wavelets, I	272	(13)
Wavelets, II	285	(27)
Quasi-Interpolation	312	(15)
Bibliography	327	(28)
Index	355	(4)
Index of Symbols	359