On the approximate form of Kluvánek's theorem
type: Item
instance of: scholarly article
title: On the approximate form of Kluvánek's theorem
zbMATH Open document ID: 1178.94167
author: M. G. Beaty
author: J. Rowland Higgins
author: M. M. Dodson
author: Simon P. Eveson
published in: Journal of Approximation Theory
publication date: 2009-12-03T00:00:00Z
review text: In earlier work of a subset of the authors [J. Fourier Anal. Appl. 13, No. 2, 187--196 (2007; Zbl 1128.94004)], a converse of Kluváneks's theorem---an extension of the classical ``Shannon'' sampling theorem to the setting of locally compact abelian groups (LCAGs)---was proved. In the present work, Kluvánek's theorem is extended to provide a formulation of the error term that arises in applying the analogue of a sinc expansion of the integer samples in the case where the analogue of the hypothesis that the square-integrable function is bandlimited is no longer valid. To state the theorem, one first needs a number of technical definitions for the LCAG context. Let \(G\) be a LCAG with dual group \(\Gamma\) having a discrete, countable subgroup \(\Lambda\) (think integers) such that the annihilator subgroup \(H=\Lambda^\bot\) of \(G\) is countable and such that the quotient \(\Gamma/\Lambda\) is compact (think torus). Let \(\Omega\) be a complete set of coset representatives of \(\Gamma/\Lambda\), called a transversal (there is always a measurable one). Its Haar measure satisfies \[ m_\Gamma (\Omega) = m_\Lambda (\{0\}) m_{\Gamma/\Lambda} (\Gamma/\Lambda) = m_\Lambda (\{0\}) /m_{H} (\{0\})<\infty. \] Denote by \(F^2(G)\) the subspace of continuous functions in \(L^2(G)\) whose group Fourier transforms are integrable. The analogue of the Paley--Wiener space of functions whose Fourier transforms are supported on an interval is the space \(\text{{PW}}_\Omega (G)\) consisting of those \(f\in F^2(G)\) whose Fourier transforms vanish off \(\Omega\). Kluvánek's theorem states that \(m_\Gamma(\Omega)\) is finite and that any \(f\in \text{{PW}}_\Omega (G)\) has the sampling representation \[ f(x) = (\mathcal{S}_H f)(x) = {1\over m_\Gamma (\Omega)} \sum_{h\in H} f(h) \chi_\Omega^{\vee} (x-h) \] converging absolutely to \(f\) pointwise and in \(L^2(G)\), and such that the \(L^2\) norm of \(f\) equals the sum of the squared moduli of the samples divided by \(m_\Gamma(\Omega)\). Th…
Mathematics Subject Classification ID: 94A20
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 94A05
zbMATH DE Number: 5641736
zbMATH Keywords: Kluvanek's theorem
zbMATH Keywords: sampling theorem
zbMATH Keywords: locally compact abelian group
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.jat.2009.02.013
OpenAlex ID: W2083713868
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DOI: 10.1016/J.JAT.2009.02.013
description: scientific article; zbMATH DE number 5641736