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On the least possible type of entire functions of order \(\rho\in(0,1)\) with positive zeros. (English. Russian original) Zbl 1214.30015

Izv. Math. 75, No. 1, 1-27 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 1, 3-28 (2011).
Summary: We find the greatest lower bound for the type of an entire function of order \( \rho \in (0,1)\) whose sequence of zeros lies on one ray and has prescribed lower and upper \( \rho\)-densities. We make a thorough study of the dependence of this extremal quantity on \( \rho\) and on properties of the distribution of zeros. The results are applied to an extremal problem on the radii of completeness of systems of exponentials.

MSC:

30D20 Entire functions of one complex variable (general theory)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D15 Special classes of entire functions of one complex variable and growth estimates
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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