Theory of Reproducing Kernels and Applications
暫譯: 再生核理論及其應用

Saitoh, Saburou, Sawano, Yoshihiro

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商品描述

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, andas 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.

商品描述(中文翻譯)

本書大幅擴展了N. Aronszajn於1950年發表的再生核一般理論,並提供了許多具體應用。在第一章中,首先介紹了許多具體的再生核,並提供詳細資訊。第二章以自足的方式呈現了再生核的一般性和全局性理論,並包含基本應用,處理了再生核希爾伯特空間之間的許多基本運算。第二章是本書的核心。第三章專注於使用再生核理論的Tikhonov正則化,並應用於有界線性算子方程的數值和實際解決方案。在第四章中,通過應用Tikhonov正則化,提出了拉普拉斯變換的數值實反演公式,其中再生核在結果中扮演了關鍵角色。第五章處理常微分方程;第六章包含各種基本偏微分方程的許多具體結果。在第七章中,介紹了典型的積分方程及其離散化方法。這些章節是第三章一般理論的應用,旨在實際和數值構造解。在第八章中,介紹了再生核的熱門主題;即,範數不等式、卷積不等式、任意矩陣的反演、反映射的表示、非線性系統的識別、取樣理論、統計學習理論和隸屬問題。還介紹了特徵函數、線性偏微分方程的初值問題和再生核之間的關係。此外,還介紹了關於廣義再生核、廣義德爾塔函數、廣義再生核希爾伯特空間以及一般積分變換理論的新基本結果。在三個附錄中,呈現了Akira Yamada深入探討非線性範數不等式中的等式問題的理論、Yamada對Opial不等式的統一和廣義不等式,以及隱函數的具體和明確的積分表示。