Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels
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.
KeywordsReproducing kernel Inequality Implicit function Singular integral Best approximation Tikhonov regularization Real inversion of Laplace transform Infinite precision method
Mathematics Subject Classification30C40 46E32 44A05 44A10 35A22 44A15 35K05 35A22 46E22
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.
- 4.Castro, L.P., Murata, K., Saitoh, S., Yamada, M.: Explicit integral representations of implicit functions. Carpath. J. Math. (accepted for publication) Google Scholar
- 6.Fujiwara, H.: High-accurate numerical method for integral equation of the first kind under multiple-precision arithmetic. Theor. Appl. Mech. Jpn. 52, 192–203 (2003) Google Scholar
- 7.Fujiwara, H., Matsuura, T., Saitoh, S., Sawano, Y.: Numerical real inversion of the Laplace transform by using a high-accuracy numerical method. In: Further Progress in Analysis, pp. 574–583. World Sci. Publ., Hackensack (2009) Google Scholar
- 11.Matkowski, J.: L p-like paranorms. In: Selected Topics in Functional Equations and Iteration Theory (Graz, 1991). Grazer Math. Ber., vol. 316, pp. 103–138 (1992). MR 94g:39010 Google Scholar
- 15.Saito, S., Saito, Y.: Yoakemae – Yocchan no Omoi (Predawn – Thoughts of Yotchan) (in Japanese). Bungeisha, Tokyo (2010) Google Scholar
- 18.Saitoh, S.: Nonlinear transforms and analyticity of functions. In: Nonlinear Mathematical Analysis and Applications, pp. 223–234. Hadronic Press, Palm Harbor (1998) Google Scholar