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Theory of Reproducing Kernels and Applications

  • Saburou Saitoh
  • Yoshihiro Sawano

Part of the Developments in Mathematics book series (DEVM, volume 44)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Saburou Saitoh, Yoshihiro Sawano
    Pages 1-63
  3. Saburou Saitoh, Yoshihiro Sawano
    Pages 65-160
  4. Saburou Saitoh, Yoshihiro Sawano
    Pages 161-196
  5. Saburou Saitoh, Yoshihiro Sawano
    Pages 197-215
  6. Saburou Saitoh, Yoshihiro Sawano
    Pages 217-230
  7. Saburou Saitoh, Yoshihiro Sawano
    Pages 231-294
  8. Saburou Saitoh, Yoshihiro Sawano
    Pages 295-319
  9. Saburou Saitoh, Yoshihiro Sawano
    Pages 321-386
  10. Back Matter
    Pages 387-452

About this book

Introduction

This book provides a large extension of the general theory of reproducing kernels published by N. Aronszajn in 1950, with many concrete applications.

In Chapter 1, many concrete reproducing kernels are first introduced with detailed information. Chapter 2 presents a general and global theory of reproducing kernels with basic applications in a self-contained way. Many fundamental operations among reproducing kernel Hilbert spaces are dealt with. Chapter 2 is the heart of this book.
Chapter 3 is devoted to the Tikhonov regularization using the theory of reproducing kernels with applications to numerical and practical solutions of bounded linear operator equations.

In Chapter 4, the numerical real inversion formulas of the Laplace transform are presented by applying the Tikhonov regularization, where the reproducing kernels play a key role in the results.
Chapter 5 deals with ordinary differential equations; Chapter 6 includes many concrete results for various fundamental partial differential equations. In Chapter 7, typical integral equations are presented with discretization methods. These chapters are applications of the general theories of Chapter 3 with the purpose of practical and numerical constructions of the solutions.

In Chapter 8, hot topics on reproducing kernels are presented; namely, norm inequalities, convolution inequalities, inversion of an arbitrary matrix, representations of inverse mappings, identifications of nonlinear systems, sampling theory, statistical learning theory and membership problems. Relationships among eigen-functions, initial value problems for linear partial differential equations, and reproducing kernels are also presented. Further, new fundamental results on generalized reproducing kernels, generalized delta functions, generalized reproducing kernel Hilbert spaces, and as well, a general integral transform theory are introduced.

In three Appendices, the deep theory of Akira Yamada discussing the equality problems in nonlinear norm inequalities, Yamada's unified and generalized inequalities for Opial's inequalities and the concrete and explicit integral representation of the implicit functions are presented.

Keywords

reproducing kernel inverse problem Hilbert space operator equation Tikhonov regularization (discretization) MSC: 30C40,44A05,35A22,30B40,45A05,41A50,46E22

Authors and affiliations

  • Saburou Saitoh
    • 1
  • Yoshihiro Sawano
    • 2
  1. 1.Gunma UniversityKiryuJapan
  2. 2.Department of Mathematics and Information ScienceTokyo Metropolitan UniversityHachiojiJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-10-0530-5
  • Copyright Information Springer Science+Business Media Singapore 2016
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-10-0529-9
  • Online ISBN 978-981-10-0530-5
  • Series Print ISSN 1389-2177
  • Series Online ISSN 2197-795X
  • Buy this book on publisher's site
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