Abstract
I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Γ/Λ
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Beaty, M., Dodson, M. & Eveson, S. A Converse to Kluvánek’s Theorem. J Fourier Anal Appl 13, 187–196 (2007). https://doi.org/10.1007/s00041-006-6025-x
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DOI: https://doi.org/10.1007/s00041-006-6025-x