A converse to Kluvánek’s theorem

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