再生核研究所

Theory of Reproducing Kernels and Applications

1st ed. 2016, Approx. 420 p. 10 illus. 
Hardcover
ISBN 978-981-10-0529-9

* Presents a unified theory of reproducing kernels that is fundamental,
beautiful and widely applicable in mathematicsDeals 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

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.