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
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