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
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
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Beesack, P.R.: Elementary proofs of some Opial-type integral inequalities. J. Anal. Math. 36(1979), 1–14 (1980). MR 81g:26006
Beesack, P.R., Das, K.M.: Extensions of Opial’s inequality. Pac. J. Math. 26, 215–232 (1968). MR 39:385
Calvert, J.: Some generalizations of Opial’s inequality. Proc. Am. Math. Soc. 18, 72–75 (1967). MR 34:4433
Castro, L.P., Murata, K., Saitoh, S., Yamada, M.: Explicit integral representations of implicit functions. Carpath. J. Math. (accepted for publication)
Castro, L.P., Saitoh, S., Sawano, Y., Simões, A.M.: General inhomogeneous discrete linear partial differential equations with constant coefficients on the whole spaces. Complex Anal. Oper. Theory 6, 307–324 (2012)
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)
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)
Fujiwara, H.: Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetic. In: Progress in Analysis and its Applications, Proceedings of the 7th International ISAAC Congress, pp. 289–295. World Scientific, Singapore (2010)
Godunova, E.K., Levin, V.I.: An inequality of Maroni (in Russian). Mat. Zametki 2, 221–224 (1967). MR 36:333
Itou, H., Saitoh, S.: Analytical and numerical solutions of linear singular integral equations. Int. J. Appl. Math. Stat. 12, 76–89 (2007)
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
Matsuura, T., Saitoh, S.: Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces. Appl. Anal. 85, 901–915 (2006)
Mitrinović, D.S.: Analytic Inequalities, Die Grundlehren der mathematischen Wissenschaften, vol. 165. Springer-Verlag, New York–Berlin (1970) xii+400 pp. MR 43:448
Opial, Z.: Sur une inégalité. Ann. Pol. Math. 8, 29–32 (1960). MR 22:3772
Saito, S., Saito, Y.: Yoakemae – Yocchan no Omoi (Predawn – Thoughts of Yotchan) (in Japanese). Bungeisha, Tokyo (2010)
Saitoh, S.: Theory of Reproducing Kernels and Its Applications. Pitman Research Notes in Mathematics Series, vol. 189. Longman Scientific & Technical, Harlow (1988) x+157 pp. MR 90f:46045
Saitoh, S.: Natural norm inequalities in nonlinear transforms. In: General Inequalities, 7 (Oberwolfach, 1995), Internat. Ser. Numer. Math., vol. 123, pp. 39–52. Birkhaüser, Basel (1997). MR 98j:46023
Saitoh, S.: Nonlinear transforms and analyticity of functions. In: Nonlinear Mathematical Analysis and Applications, pp. 223–234. Hadronic Press, Palm Harbor (1998)
Saitoh, S.: Integral Transforms, Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series, vol. 369. Longman, Harlow (1997) xiv+280 pp. MR 98k:46041
Saitoh, S.: Approximate real inversion formulas of the Laplace transform. Far East J. Math. Sci. 11, 53–64 (2003)
Sawano, Y., Yamada, M., Saitoh, S.: Singular integrals and natural regularizations. Math. Inequal. Appl. 13, 289–303 (2010)
Yamada, A.: Saitoh’s inequality and Opial’s inequality. Math. Inequal. Appl. 14(3), 523–528 (2011)
Yamada, M., Saitoh, S.: Identifications of non-linear systems. J. Comput. Math. Optim. 4(1), 47–60 (2008). MR 2009b:93051
Yamada, M., Saitoh, S.: Explicit and direct representations of the solutions of nonlinear simultaneous equations. In: Progress in Analysis and its Applications, Proceedings of the 7th International ISAAC Congress, pp. 372–378. World Scientific, Singapore (2010)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to the Memory of Wolfgang Walter
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Castro, L.P., Fujiwara, H., Saitoh, S., Sawano, Y., Yamada, A., Yamada, M. (2012). Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0249-9_6
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0248-2
Online ISBN: 978-3-0348-0249-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)