Theory of Reproducing Kernels and Applications

1. Front Matter

   Pages i-xviii

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2. Definitions and Examples of Reproducing Kernel Hilbert Spaces

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 1-63

3. Fundamental Properties of RKHS

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 65-160

4. Moore Penrose Generalized Inverses and Tikhonov Regularization

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 161-196

5. Real Inversion Formulas of the Laplace Transform

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 197-215

6. Applications to Ordinary Differential Equations

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 217-230

7. Applications to Partial Differential Equations

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 231-294

8. Applications to Integral Equations

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 295-319

9. Special Topics on Reproducing Kernels

   • Saburou Saitoh, Yoshihiro Sawano

   Pages 321-386

10. Back Matter

    Pages 387-452

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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 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.

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

• Gunma University, Kiryu, Japan

  Saburou Saitoh

• Department of Mathematics and Information Science, Tokyo Metropolitan University, Hachioji, Japan

  Yoshihiro Sawano

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