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.
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