Abstract
We would like to propose a new method in view to catch smoothing properties and analyticity of functions by computers. Of course, in the strict sense, such goal is impossible. However, we would like to propose some practical method that may be applied for many concrete cases for some good functions (but not for bad functions, in a sense). Therefore, this may be viewed as a procedure proposal which includes numerical experiments for the just mentioned challenge and within a new method.
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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, p. 52 Springer, New York. http://www.springer.com/mathematics/analysis/book/978-1-4939-1105-9
Fujiwara, H.: Applications of reproducing kernel spaces to real inversions of the Laplace transform. RIMS Koukyuuroku 1618, 188–209 (2008)
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, Hackensack (2010)
Fujiwara, H.: Exflib: multiple-precision arithmetic library (2005). http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib
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 Scientific, Hackensack (2009)
Macintyre, A.J., Rogosinski, W.W.: Extremum problems in the theory of analytic functions. Acta Math. 82, 275–325 (1950)
Riesz, F.: Über Potenzreihen mit vorgeschriebenen Anfangsgliedern. Acta Math. 42, 145–171 (1920)
Saitoh, S.: Integral transforms, reproducing kernels and their applications. Pitman Research Notes in Mathematical 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. American Mathematical Society Translations Series 2, vol. 230. American Mathematical Society, Providence (2010)
Tan, L., Qian, T.: Backward shift invariant subspaces with applications to band preserving and phase retrieval problems (manuscript)
Acknowledgements
This work was supported in part by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT), within project PEst-OE/MAT/ UI4106/2014. The fourth named author is supported in part by the Grant-in-Aid for the Scientific Research (C) (2) (No. 24540113).
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Castro, L.P., Fujiwara, H., Qian, T., Saitoh, S. (2014). How to Catch Smoothing Properties and Analyticity of Functions by Computers?. In: Pardalos, P., Rassias, T. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1124-0_4
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DOI: https://doi.org/10.1007/978-1-4939-1124-0_4
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