Theory of Reproducing Kernels and Applications

ISBN-13: 9789811005299 / Angielski / Twarda / 2016 / 452 str.

Saburou Saitoh; Yoshihiro Sawano

Spis treści

Preface.- Part I Preliminaries: Integration.- Fundamental properties of integrals.- Normed spaces and Banach spaces.- Hilbert spaces.- Fourier integrals.- Part II Definitions and examples of reproducing kernel Hilbert spaces: What is an RKHS?.- RKHS of Sobolev type.- Paley‐Wiener reproducing kernels.- RKHS and complex analysis on C.- RKHS in the spaces of polynomials.- Dirac’s delta function, Green’s functions and reproducing kernels.- Part III Fundamental properties of RKHS: Basic properties of RKHS.- Operations of more than one reproducing kernel Hilbert space.- Construction of reproducing kernel Hilbert spaces.- RKHS and linear mappings.- Part IV Moore‐Penrose generalized inverses and Tikhonov regularization: The best approximations and the Moore‐Penrose generalized inverses.- Spectral analysis and the Tikhonov regularization.- Part V Real inversion formula of the Laplace transform: Real inversion formulas of the Laplace transform.- Part VI Applications to ordinary differential equations: By the use of the Tikhonov regularization.- Discrete ordinary linear di.erential equations.- Part VII Applications to partial di.erential equations: Poisson’s equation.- Laplace’s equation.- Heat equation.- Wave equation.- General inhomogeneous PDEs on the whole spaces.- PDEs and inverse problems, a general approach.- Part VIII Applications to integral equations: Singular integral equations.- Convolution equations and general fractional functions.- Convolution integral equations.- Integral equations with the mixed Toeplitz‐‐Hankel kernel.- Part IX Special topics on reproducing kernels: Norm inequalities.- Convolution norm inequalities.- Inequalities for Gram matrices.- Inversion for any matrix by the Tikhonov regularization.- Representation of inverse functions.- Identifications of non‐linear systems.- Sampling theory.- Error and convergence rate estimates in statistical learning theory.- Membership problems for RKHSs.- Part X Appendices: Equality conditions for norm inequalities derived from the theory of reproducing kernels.- Generalizations of Opial’s inequality.- Explicit integral representations of implicit functions.-Overview on the Theory of Reproducing Kernels.- References.- Index.

Blurb

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.