Overview
- Presents a unified theory of reproducing kernels that is fundamental, beautiful and widely applicable in mathematics
- Deals with the new discretizations and the Tikhonov regularization for practical constructions of the solutions by computers in analysis
- Introduces many global, up-to-date topics of general interest from the general theory of N. Aronszajn
Part of the book series: Developments in Mathematics (DEVM, volume 44)
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Table of contents (8 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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, andas well, a general integral transform theory are introduced.
In three Appendices, the deep theory ofAkira 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.
Authors and Affiliations
Bibliographic Information
Book Title: Theory of Reproducing Kernels and Applications
Authors: Saburou Saitoh, Yoshihiro Sawano
Series Title: Developments in Mathematics
DOI: https://doi.org/10.1007/978-981-10-0530-5
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media Singapore 2016
Hardcover ISBN: 978-981-10-0529-9Published: 27 October 2016
Softcover ISBN: 978-981-10-9184-1Published: 22 April 2018
eBook ISBN: 978-981-10-0530-5Published: 14 October 2016
Series ISSN: 1389-2177
Series E-ISSN: 2197-795X
Edition Number: 1
Number of Pages: XVIII, 452
Number of Illustrations: 1 b/w illustrations
Topics: Functional Analysis, Fourier Analysis, Functions of a Complex Variable, Partial Differential Equations