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Theory of Reproducing Kernels and Applications

Part of the Developments in Mathematics Book Series (volume 44)

DOI: 10.1007/978-981-10-0530-5

ISBN: 978-981-10-0529-9

Publisher: Springer Singapore - Springer Journals

Oct 15, 2016

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Chapter 1Definitions and Examples of Reproducing Kernel Hilbert Spaces

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 1 –63

[In Chap. 1 we introduce many examples of reproducing kernel Hilbert spaces (RKHSs for short). The reader may skip Chap. 1 and later go back to it if needed. We will consider RKHSs of Sobolev type, of Paley Wiener type, on the complex plane \documentclass[12pt]{minimal} \usepackage{amsmath}...
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Chapter 2Fundamental Properties of RKHS

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 65 –160

[In Chap. 2, which is at the heart of this book, we develop the general theory of reproducing kernels and its fundamental properties. First, we establish that any positive definite kernel corresponds uniquely to an RKHS. Then we discuss operations for RKHS. These operations will be fundamental...
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Chapter 3Moore Penrose Generalized Inverses and Tikhonov Regularization

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 161 –196

[By applying the general theory of reproducing kernels, we will consider the practical constructions of approximate solutions for bounded linear operator equations.]
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Chapter 4Real Inversion Formulas of the Laplace Transform

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 197 –215

[As stated in the preface, one of our strong motivations for writing this book is given by the historical success of the numerical and real inversion formulas of the Laplace transform which is a famous typical ill-posed and very difficult problem. In this chapter, we will see their mathematical...
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Chapter 5Applications to Ordinary Differential Equations

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 217 –230

[We will give concrete representation formulas of the approximate solutions for linear ordinary differential equations following the Tikhonov regularization and using a finite number of data.]
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Chapter 6Applications to Partial Differential Equations

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 231 –294

[In this chapter, we gather concrete results on typical linear partial differential equations using Tikhonov regularization and a finite number of data.]
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Chapter 7Applications to Integral Equations

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 295 –319

[Singular integral equations are presently encountered in a wide range of mathematical models, for instance in acoustics, fluid dynamics, elasticity and fracture mechanics. Together with these models, a variety of methods and applications for these integral equations has been developed. For...
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Chapter 8Special Topics on Reproducing Kernels

Saitoh, Saburou; Sawano, Yoshihiro

Pages: 321 –386

[In this chapter, we will gather simply several special hot and important topics from the viewpoint of the general theory of reproducing kernels.]

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