Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels

Castro, L. P.; Fujiwara, Hiroshi; Saitoh, Saburou; Sawano, Yoshihiro; Yamada, Akira; Yamada, Masumi

doi:10.1007/978-3-0348-0249-9_6

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…For up-to-date versions of the Tikhonov regularization by using the theory of reproducing kernels, and some concrete classes of operators, see [1,2,3,5,6,7,8]. Cf.…”

Section: Known Backgroundmentioning

confidence: 99%

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

Castro

Saitoh²,

Yamada

2016

Complex Anal. Oper. Theory

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We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert-Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.

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“…For up-to-date versions of the Tikhonov regularization by using the theory of reproducing kernels, and some concrete classes of operators, see [1,2,3,5,6,7,8]. Cf.…”

Section: Known Backgroundmentioning

confidence: 99%

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

Castro

Saitoh²,

Yamada

2016

Complex Anal. Oper. Theory

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“…For up-to-date versions of the Tikhonov regularization using the theory of reproducing kernels, see [7,9].…”

Section: Theorem 6 Let L : H K → H Be a Bounded Linear Operator Andmentioning

confidence: 99%

Aveiro Discretization Method in Mathematics: A New Discretization Principle

Castro

Fujiwara

Rodrigues

et al. 2014

Mathematics Without Boundaries

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We found a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method is here performed by considering several distinct classes of problems to which the method is applied. In fact, one of the advantages of the present method -in comparison to other well-known and well established methods -is its global nature and no need of special or very particular data conditions. Numerical experiments have been made, and consequent results are here exhibited. Due to the powerful results which arise from the application of the present method, we consider that this method has everything to become one of the next generation methods of solving general analytical problems by using computers. In particular, we would like to point out that we will be able to solve very global linear partial differential equations satisfying very general boundary conditions or initial values (and in a somehow independent way of the boundary and domain). Furthermore, we will be able to give an ultimate sampling theory and an ultimate realization of the consequent general reproducing kernel Hilbert spaces. The general theory is here presented in a constructive way, and containing some related historical and concrete examples.

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A Reproducing Kernel Theory with Some General Applications

Saitoh¹

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels

Cited by 4 publications

References 9 publications

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

Aveiro Discretization Method in Mathematics: A New Discretization Principle

A Reproducing Kernel Theory with Some General Applications

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