Estimate of the Upper Density of Gabor System

Isaev, K. P. , Fazullin, Z. Yu.  , Yulmukhametov, R. S.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 4.

Abstract:
In [1] it was shown that the upper density of a discrete set \(\Lambda \) for which the Gabor system \(G_\Lambda \) is complete in the space \(L^2(\Bbb R)\) cannot be less than \(\frac 1{3\pi }\). It is also known from earlier studies that with a regular distribution of indicators, the upper density is not less than \(\frac{2}{\pi} \). In this paper, we refine the estimate in the absence of the regularity condition for the distribution: the upper density of a discrete set \(\Lambda \) for which the Gabor system \(G_\Lambda\) is complete in the space \(L^2(\Bbb R)\) cannot be less than \(\frac {\sqrt 3}{4\pi }\). Improvement of the estimates is achieved by a more methodical application of symmetrization of this set of indicators of the Gabor system using the known effect of reducing the growth of the modulus of an entire function with a more symmetrical arrangement of its zeros. The possibility of improving the obtained estimate within the proposed method is also discussed using specific examples.

Keywords: completeness, Gabor system, frame, density, fock space, uniqueness set

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