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Shannon–Whittaker–Kotel’nikov’s theorem generalized revisited

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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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Correspondence to Juan L. G. Guirao.

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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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