Theory of reproducing kernels and applications. (2016)

• Record Type: Book
• Further Information: Note: Saburou Saitoh, Yoshihiro Sawano.
• Authors: Saitoh, Saburou, 1944-; Sawano, Yoshihiro
• Contents: 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.
• Publisher Details: Singapore : Springer
• Publication Date: 2016
• Copyright Date: 2016
• Extent: 1 online resource (452 pages)
• Subjects: 515.733; Kernel functions; Hilbert space
• Languages: English
• ISBNs: 9789811005305
• Related ISBNs: 9789811005299
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.