Abstract
We give a short survey of a general discretization method based on the theory of reproducing kernels. We believe our method will become the next generation method for solving analytical problems by computers. For example, for solving linear PDEs with general boundary or initial value conditions, independently of the domains. Furthermore, we give an ultimate sampling formula and a realization of reproducing kernel Hilbert spaces.
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References
L.P. Castro, H. Fujiwara, M.M. Rodrigues, S. Saitoh, V.K. Tuan, Aveiro discretization method in mathematics: a new discretization principle, in Mathematics Without Boundaries: Surveys in Pure Mathematics, ed. by P. Pardalos, T.M. Rassias (Springer, New York, to appear). 52 pp.
L.P. Castro, H. Fujiwara, T. Qian, S. Saitoh, How to catch smoothing properties and analyticity of functions by computers? in Mathematics Without Boundaries: Surveys in Pure Mathematics, ed. by P. Pardalos, T.M. Rassias (Springer, New York, to appear). 15 pp.
L.P. Castro, S. Saitoh, Y. Sawano, N.M. Tuan, Approximate solutions of arbitrary linear ordinary differential equations (submitted for publication)
H. Fujiwara, Applications of reproducing kernel spaces to real inversions of the Laplace transform. RIMS Kokyuroku 1618, 188–209 (2008)
H. Fujiwara, T. Matsuura, S. Saitoh, Y. Sawano, Numerical real inversion of the Laplace transform by using a high-accuracy numerical method, in Further Progress in Analysis (World Scientific, Hackensack, 2009), pp. 574–583
H. Fujiwara, Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetric, in Proceedings of the 7th International ISAAC Congress. Progress in Analysis and Its Applications (World Scientific, Singapore, 2010), pp. 289–295
S. Saitoh, Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74–78 (1983)
S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications. Pitman Res. Notes in Math. Series, vol. 369 (Addison-Wesley/Longman, Reading/Harlow, 1997)
S. Saitoh, Theory of reproducing kernels: applications to approximate solutions of bounded linear operator functions on Hilbert spaces, in Am. Math. Soc. Transl. Ser. 2, vol. 230 (Am. Math. Soc., Providence, 2010), pp. 107–134
Acknowledgements
This work was supported in part by the CIDMA—Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology, within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690, as well as by the Grant-in-Aid for the Scientific Research (C)(2) (No. 24540113).
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Castro, L.P., Fujiwara, H., Rodrigues, M.M., Saitoh, S., Tuan, V.K. (2015). Reproducing Kernels and Discretization. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_61
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DOI: https://doi.org/10.1007/978-3-319-12577-0_61
Publisher Name: Birkhäuser, Cham
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