Density of zeros of the Cartwright class functions and the Helson--Szeg\"o type condition

DOI10.1134/S0001434623010194zbMATH Open1519.46019arXiv2208.11350MaRDI QIDQ6408659

Sergei Avdonin, S. A. Ivanov

Publication date: 24 August 2022

Abstract: B.,Ya.,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's result to a more general class of entire functions $F(z)$ with zeros in a strip $0<qleqImzleqQ,$ such that $|F(x){|}^2$ satisfies the Helson--Szeg"o condition. Moreover, we demonstrate that instead of the last condition one can require that $log|F(x)|$ belongs to the BMO class.

Mathematics Subject Classification ID

Spaces of measurable functions ((L^p)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) (46E30) Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces (46B15) Completeness problems, closure of a system of functions of one complex variable (30B60)

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