Abstract
In Antuña et al. (MATCH Commun Math Chem 73:385–396, 2015) was proved that if \({{\lambda }}= \{ \lambda _k \}_{k \in {\mathbb {Z}}}\) is a bounded sequence of positive real numbers holding the property \({\sum _{\begin{array}{c} k \in {\mathbb {Z}}\\ k \ne 0 \end{array}}^{} \left| {{\log \lambda _k} \over k} \right| < \infty }\) then the function \({\sigma _{\lambda }(t) := \prod _{k \in {\mathbb {Z}}} \lambda _k^{\text {sinc}(t-k)}}\) holds the Shannon–Whittaker–Kotel’nikov’s theorem generalized (SWKTG) and it can be recomposed in the way \({\sigma _{\lambda }(t)=\lim _{n \rightarrow \infty } {\left( \sum _{k \in {\mathbb {Z}}} \lambda _k^{1 \over n} \text {sinc}(t-k) \right) }^n}\) for every \(t \in {\mathbb {R}}\). The aim of the present work is to analyze the algebraic structure of the set of sequences of positive real numbers holding \({\sum _{\begin{array}{c} k \in {\mathbb {Z}}\\ k \ne 0 \end{array}}^{} \left| {{\log \lambda _k} \over k} \right| < \infty }\). It will allow to apply SWKTG in a more effective way in its many applications, in particular to the chemical ones.
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References
A. Antuña, J.L.G. Guirao, M.A. López, An asymptotic sampling recomposition theorem for Gaussian signals. Mediterr. J. Math. 8, 349–367 (2011)
A. Antuña, J.L.G. Guirao, M.A. López, Pseudo-radioactive decomposition through a generalized Shannon’s recomposition theorem. MATCH Commun. Math. Comput. Chem. 72(2), 403–410 (2014)
A. Antuña, J.L.G. Guirao, M.A. López, Shannon-Whittaker-Kotel’nikov’s theorem generalized. MATCH Commun. Math. Comput. Chem. 73, 385–396 (2015)
P.L. Butzer, S. Ries, R.L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series. J. Approx. Theory 50, 25–39 (1987)
P.L. Butzer, R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Rev. 34(4), 40–53 (1992)
M.T. de Bustos, J.L.G. Guirao, J. Vigo-Aguilar, Descomposition of pseudo-radioactive chemical products with a mathematical approach. J. Math. Chem. 52(4), 1059–1065 (2014)
J.A. Gubner, A new series for approximating Voight functions. J. Phys. A: Math. 27, L745–L749 (1994)
J.L.G. Guirao, M.T. de Bustos, Dynamics of pseudo-radioactive chemical products via sampling theory. J. Math. Chem. 50(2), 374–378 (2012)
J.R. Higgins, Sampling Theory in Fourier and Signals Analysis: Foundations (Oxford Univ. Press, Oxford, 1996)
S.M. Hosamani, Correlation of domination parameters with physicochemical properties of octane isomers. Appl. Math. Nonlinear Sci. 1(2), 345–352 (2018)
D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960)
C.E. Shannon, Communication in the presence of noise. Proc. IRE 137, 10–21 (1949)
E.T. Whittaker, On the functions which are represented by the expansions of the interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)
A.I. Zayed, Advances in Shannon’s Sampling Theory (CRC Press, Boca Raton, 1993)
B. Zhoa, H. Wu, Pharmacological characteristics analysis of two molecular structures. Appl. Math. Nonlinear Sci. 2(1), 93–110 (2017)
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Antuña, A., Guirao, J.L.G. & López, M.A. Shannon–Whittaker–Kotel’nikov’s theorem generalized revisited. J Math Chem 58, 893–905 (2020). https://doi.org/10.1007/s10910-019-01037-w
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DOI: https://doi.org/10.1007/s10910-019-01037-w