A Converse to Kluvánek’s Theorem

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 ofKluvánek'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 / .

On the approximate form of Kluvánek's theorem

“…In a converse result, one or both of these equations can be assumed to hold for L 2 (G) ∩ C(G) or the class C A or for a subset. A converse (Theorem 4) has been proved in [6] under the hypothesis that there exists some function f represented by (25) but its Fourier transform f ∧ must essentially contain A (the representation implies essential equality). Both (25) and (26) are needed as hypotheses (note that (26), the square-summability hypothesis, was overlooked).…”

“…The notation for this set has been changed to C A since the notation Section A used in other related papers, such as [5] and [6], could be confused with the sampling operator notation S w used in (7) and in [13,15]. Kluvánek …”

“…Details about further definitions and notation from abstract harmonic analysis can be found in [3,17,24,25,36,38]. A brief account relevant to this paper is in [6]. Throughout G will be a locally compact abelian group with translation invariant Haar measure m G , Haar integral…”

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

“…accounts are in [6,7,28]. The website [65] of the Numerical Harmonic Analysis Group (NuHAG) at the University of Vienna is a comprehensive and up-to-date source of information about all aspects of sampling theory.…”

“…An asymptotic formula needs more information [7,Thms. 5,6]. A norm result for f corresponding to Kluvánek's theorem is not known, although putting f = g in the generalized or approximate Parseval formula (1.1) suggests that modulo some summability conditions…”