Abstract:
In the work we find the minimal value that can be taken by the type of an entire function of order $\rho\in(0,1)$ with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than $\pi$. The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A. Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.
Keywords:
type of an entire function, upper and lower density of zeroes, uniqueness theorem, completeness of exponential system.
Citation:
V. B. Sherstyukov, “Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a prescribed density”, Ufa Math. J., 8:1 (2016), 108–120
\Bibitem{She16}
\by V.~B.~Sherstyukov
\paper Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a~prescribed density
\jour Ufa Math. J.
\yr 2016
\vol 8
\issue 1
\pages 108--120
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\crossref{https://doi.org/10.13108/2016-8-1-108}
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Linking options:
https://www.mathnet.ru/eng/ufa318
https://doi.org/10.13108/2016-8-1-108
https://www.mathnet.ru/eng/ufa/v8/i1/p113
This publication is cited in the following 3 articles:
G. G. Braichev, V. B. Sherstyukov, “Otsenki indikatorov tseloi funktsii s otritsatelnymi kornyami”, Vladikavk. matem. zhurn., 22:3 (2020), 30–46
V. B. Sherstyukov, “Asymptotic properties of entire functions with given laws of distribution of zeros”, J. Math. Sci. (N. Y.), 257:2 (2021), 246–272
G. G. Braichev, V. B. Sherstyukov, “Sharp bounds for asymptotic characteristics of growth of entire functions with zeros on given sets”, J. Math. Sci., 250:3 (2020), 419–453
Citation in format AMSBIB
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