Sampling and Approximation in Shift Invariant Subspaces of L 2 ( ℝ ) $$L_2(\mathbb {R})$$

Nikolaos Atreas ()
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Nikolaos Atreas: Aristotle University of Thessaloniki

A chapter in Harmonic Analysis and Applications, 2021, pp 1-19 from Springer

Abstract: Abstract Let ϕ be a continuous function in L 2 ( ℝ ) $$L_2(\mathbb {R})$$ with a certain decay at infinity and a non-vanishing property in a neighborhood of the origin for the periodization of its Fourier transform ϕ ^ $$\widehat {\phi }$$ . Under the above assumptions on ϕ, we derive uniform and non-uniform sampling expansions in shift invariant spaces V ϕ ⊂ L 2 ( ℝ ) $$V_{\phi } \subset L_2(\mathbb {R})$$ . We also produce local (finite) sampling formulas, approximating elements of V ϕ in bounded intervals of ℝ $$\mathbb {R}$$ , and we provide estimates for the corresponding approximation error, namely, the truncation error. Our main tools to obtain these results are the finite section method and the Wiener’s lemma for operator algebras.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-61887-2_1

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DOI: 10.1007/978-3-030-61887-2_1

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