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On best localized bandlimited functions

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Abstract

Let, for σ > 0, \(\mathcal{B}_\sigma \) be the set of complex functions f ∈ L 1 (ℝ) with the Fourier transforms \(\hat f(x)\smallint _\mathbb{R} e^{ - 2\pi ixt} f(t)dt\) vanishing outside the interval [−σ; σ]. In this paper, we study the problem of the best approximation of the Dirac function δ (which has the Fourier transform with widest support supp\((\hat \delta ) = \mathbb{R}\)) by functions \(f \in \mathcal{B}_\sigma \). More precisely, we consider the quantity inf\(\{ \sum _{n \in \mathbb{Z}} |f(n)|:f \in \mathcal{B}_\sigma ,f(0) = 1\} \) and its extremal functions \(f \in \mathcal{B}_\sigma \).

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, 08663, Vilnius, Lithuania

    S. Norvidas

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  1. S. Norvidas
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Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 548–564, October–December, 2006.

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Norvidas, S. On best localized bandlimited functions. Lith Math J 46, 446–458 (2006). https://doi.org/10.1007/s10986-006-0041-z

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  • DOI: https://doi.org/10.1007/s10986-006-0041-z

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