Abstract
We would like to propose a new method for the sampling theory which represents the functions by a finite number of point data in a very general reproducing kernel Hilbert space function space. The result may be looked as an ultimate sampling theorem in a reasonable sense. We shall give numerical experiments also as its evidences.
In Honor of Constantin Carathéodory
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Castro, L.P., Fujiwara, H., Rodrigues, M.M., Saitoh, S.: A new discretization method by means of reproducing kernels. In: Son, L.H., Tutscheke, W. (eds.) Interactions Between Real and Complex Analysis, pp. 185–223. Science and Technology Publication House, Hanoi (2012)
Castro, L.P., Fujiwara, H., Rodrigues, M.M., Saitoh, S., Tuan, V.K.: Aveiro discretization method in mathematics: a new discretization principle. In: Pardalos, P., Rassias, T.M. (eds.) Mathematics without Boundaries: Surveys in Pure Mathematics, pp. 37–92. Springer, New York (2014)
Fujiwara, H.: Exflib: multiple-precision arithmetic library for scientific computation. http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib (2005)
Fujiwara, H.: Numerical real inversions of the Laplace transform and their applications. RIMS Koukyuuroku 1618, 192–209 (2008)
Fujiwara, H.: Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetric. In: Progress in Analysis and its Applications, Proceedings of the 7th International ISAAC Congress, pp. 289–295. World Scientific, Singapore (2010)
Fujiwara, H., Matsuura, T., Saitoh, S., Sawano, Y.: Numerical real inversion of the Laplace transform by using a high-accuracy numerical method. Further Progress in Analysis, pp. 574–583. World Scientific Publishing, Hackensack (2009)
Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Clarendon Press, Oxford (1996)
Higgins, J.R., Stens, R.L. (ed.): Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Clarendon Press, Oxford (1999)
Jerri, A.J.: The Shannon sampling theorem—its various extensions and applications: a tutorial review. Proc. IEEE 65, 1565–1596 (1977)
Mo, Y., Qian, T.: Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem. Appl. Math. Comput. 245, 509–519 (2014)
Saitoh, S.: Integral Transforms, Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series, vol. 369. Addison Wesley Longman, Harlow (1997)
Saitoh, S.: Theory of reproducing kernels: applications to approximate solutions of bounded linear operator functions on Hilbert spaces. Am. Math. Soc. Transl. Ser. 2 230, 107–134 (2010)
Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, vol. 20. Springer, New York (1993)
Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998).
Acknowledgements
The first and the second author are supported by JSPS KAKENHI Grant Number (C)(No. 26400198), (C)(No. 24540113), respectively.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fujiwara, H., Saitoh, S. (2016). The General Sampling Theory by Using Reproducing Kernels. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-31317-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31315-3
Online ISBN: 978-3-319-31317-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)