Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels

  • Luís P. Castro
  • Hiroshi Fujiwara
  • Saburou Saitoh
  • Yoshihiro Sawano
  • Akira Yamada
  • Masato Yamada
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)

Abstract

First of all, we will be concentrated in some particular but very important inequalities. Namely, for a real-valued absolutely continuous function on [0,1], satisfying f(0)=0 and \(\int_{0}^{1}f'(x)^{2}\,dx<1\), we have, by using the theory of reproducing kernels
$$\int_0^1\left(\frac{f(x)}{1-f(x)}\right)^{\prime\,2}(1-x)^2\,dx \le\frac{\int_0^1f^{\prime\,2}(x)\,dx}{1-\int_0^1f^{\prime\,2}(x)\,dx}. $$
A. Yamada gave a direct proof for this inequality with a generalization and, as an application, he unified the famous Opial inequality and its generalizations.

Meanwhile, we gave some explicit representations of the solutions of nonlinear simultaneous equations and of the explicit functions in the implicit function theory by using singular integrals. In addition, we derived estimate inequalities for the consequent regularizations of singular integrals.

Our main purpose in this paper is to introduce our method of constructing approximate and numerical solutions of bounded linear operator equations on reproducing kernel Hilbert spaces by using the Tikhonov regularization. In view of this, for the error estimates of the solutions, we will need the inequalities for the approximate solutions. As a typical example, we shall present our new numerical and real inversion formulas of the Laplace transform whose problems are famous as typical ill-posed and difficult ones. In fact, for this matter, a software realizing the corresponding formulas in computers is now included in a present request for international patent. Here, we will be able to see a great computer power of H. Fujiwara with infinite precision algorithms in connection with the error estimates.

Keywords

Reproducing kernel Inequality Implicit function Singular integral Best approximation Tikhonov regularization Real inversion of Laplace transform Infinite precision method 

Mathematics Subject Classification

30C40 46E32 44A05 44A10 35A22 44A15 35K05 35A22 46E22 

Notes

Acknowledgements

S. Saitoh is supported in part by the Grant-in-Aid for the Scientific Research (C)(2) (No. 21540111) from the Japan Society for the Promotion Science and Y. Sawano was supported by Grant-in-Aid for Young Scientists (B) (No. 21740104) Japan Society for the Promotion of Science. L. Castro and S. Saitoh are supported in part by Center for Research and Development in Mathematics and Applications, University of Aveiro, Portugal, through FCT—Portuguese Foundation for Science and Technology.

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Luís P. Castro
    • 1
  • Hiroshi Fujiwara
    • 2
  • Saburou Saitoh
    • 1
  • Yoshihiro Sawano
    • 3
  • Akira Yamada
    • 4
  • Masato Yamada
    • 5
  1. 1.Center for R&D in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Graduate School of InformaticsKyoto UniversityKyotoJapan
  3. 3.Department of MathematicsKyoto UniversityKyotoJapan
  4. 4.Department of MathematicsTokyo Gakugei UniversityKoganei-shiJapan
  5. 5.University Education CenterGunma UniversityMaebashi CityJapan

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