Frequency-domain bounds for nonnegative, unsharply band-limited functions

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Abstract

In this paper we present a technique for proving bounds of the Boas-Kac-Lukosz type for unsharply restricted functions with nonnegative Fourier transforms. Hence we consider functions F(x) ≥ 0, the Fourier transform f(u) of which satisfies |f(u)| ≤ ε for all u in a subset of (-∞,-1] ⋃ [1,∞), and are interested in bounds on |f(u)| for |u| ≤ 1. This technique gives rise to several "epsilonized" versions of the Boas-Kac-Lukosz bound (which deals with the case f(u) = 0, |u| ≥ 1). For instance, we find that |f(u)| ≤ L(u) + O(ε2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. The technique also turns out tobe sufficiently powerful to yield the best bound as ε ↓ 0 in various other cases with less severe restrictions on f.

Original languageEnglish
Pages (from-to)39-65
Number of pages27
JournalJournal of Fourier Analysis and Applications
Volume1
Issue number1
DOIs
Publication statusPublished - 1 Jan 1994
Externally publishedYes

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