On the approximate form of Kluvánek's theorem
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integer samples in the case where the analogue of the hypothesis that the square-integrable function is bandlimited is no longer valid.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 \(m_\Gamma(\Omega)\). The authors extend the sampling representation to an approximate projection formula \[ f(t) = (\mathcal{S}_H f)(t) +(\mathcal{R}_H f)(t) \] in which \(\mathcal{R}_H f\) satisfies \(|(\mathcal{R}_H f)(t)|\leq 2\int_{\Gamma\setminus\Omega} |\widehat{f}(\gamma)|\, dm_\Gamma (\gamma)\). Here, one assumes that \(f\in F^2(G)\) and \(f\in \ell^2(H)\) so that \(\mathcal{S}_H f\) is defined. This result is an extension of the known case of the real line. Much of the present work is dedicated to quantifying the senses in which the approximations converge to \(f\), including the case in which \(G\) has a suitable ``increasing family of transversals'' corresponding to the real line case in which \(f\) is not bandlimited, but has successive approximations by a sequence of bandlimited functions with increasing bandwidths.
scientific article; zbMATH DE number 5641736
scientific article