Integral Transforms, Reproducing Kernels and Their Applications

by Saitoh; Saburou

• ISBN13: 9780582317581
• ISBN10: 0582317584
• Format: Paperback
• Copyright: 1997-05-28
• Publisher: Chapman & Hall/

Summary

The general theories contained in the text will give rise to new ideas and methods for the natural inversion formulas for general linear mappings in the framework of Hilbert spaces containing the natural solutions for Fredholm integral equations of the first kind.

Table of Contents

Preface
Acknowledgments
1. Introduction	1	(19)
2. Reproducing kernel Hilbert spaces	20	(77)
1 Fundamental theorems of linear transforms	20	(14)
2 Elementary properties of reproducing kernel Hilbert spaces	34	(9)
3 Generations of reproducing kernel Hilbert spaces	43	(10)
4 Examples of reproducing kernels	53	(34)
5 Realizations of reproducing kernel Hilbert spaces	87	(10)
3. Isometrical identities and inversion formulas	97	(81)
1 Laplace and Fourier transforms	97	(31)
2 The heat equation	128	(15)
3 The wave equation	143	(11)
4 Analytic and harmonic functions of class L(2)	154	(14)
5 The Meyer wavelets	168	(10)
4. Applications to the approximation of functions	178	(22)
1 Best approximations by the functions in a RKHS	178	(11)
2 Approximations in reproducing kernel Hilbert spaces	189	(11)
5. Applications to analytic extension formulas and real inversion formulas for the Laplace transform	200	(18)
1 Representations of the norms in Bergman-Selberg spaces	200	(11)
2 Real inversion formulas for the Laplace transform	211	(7)
6. Applications to source inverse problems	218	(19)
1 Inverse source formulas in Poisson's equation	218	(11)
2 Inverse source formulas for Helmholtz's equation	229	(8)
Appendix 1. Applications to representations of inverse functions	237	(6)
1 A general approach	237	(2)
2 Reasonable settings	239	(1)
3 Examples	240	(3)
Appendix 2. Natural norm inequalities in nonlinear transforms	243	(16)
1 General principles	243	(5)
2 Concrete examples	248	(11)
Appendix 3. Stability of Lipschitz type in determination of initial heat distribution	259	(8)
1 Introduction and theorem	259	(3)
2 Proof of theorem	262	(5)
References	267	(1)
Books	267	(5)
Articles	272