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On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(ℝ)

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Abstract

Let ϕ be a function in the Wiener amalgam space \(\emph{W}_{\infty}(L_1)\) with a non-vanishing property in a neighborhood of the origin for its Fourier transform \(\widehat{\phi}\), \({\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}}\) be a sampling set on ℝ and \(V_\phi^{\bf \tau}\) be a closed subspace of \(L_2(\hbox{\ensuremath{\mathbb{R}}})\) containing all linear combinations of τ-translates of ϕ. In this paper we prove that every function \(f\in V_\phi^{\bf \tau}\) is uniquely determined by and stably reconstructed from the sample set \(L_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}\). As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of ϕ we provide an estimate to the corresponding error.

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Authors and Affiliations

  1. Department of Mathematics, Physics and Computational Sciences, Faculty of Engineering, Aristotle University of Thessaloniki, 54–124, Thessaloniki, Greece

    Nikolaos D. Atreas

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  1. Nikolaos D. Atreas
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Correspondence to Nikolaos D. Atreas.

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Communicated by Qiyu Sun.

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Atreas, N.D. On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(ℝ). Adv Comput Math 36, 21–38 (2012). https://doi.org/10.1007/s10444-011-9177-4

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  • DOI: https://doi.org/10.1007/s10444-011-9177-4

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