Distributions of zeros and masses of entire and

B. N. Khabibullin

Izvestiya. Mathematics , Tome 88 (2024) no. 1, pp. 133-193

Résumé

Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex plane $\mathbb C$. The following problem dates back to F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$, for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there exists an entire function $f\neq 0$ of exponential type that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis? The classical Malliavin–Rubel theorem of the early 1960s completely solves this problem for “positive” $\mathrm Z$ and $\mathrm W$ (which lie only on the positive semiaxis). Several generalizations of this criterion were established by the author of the present paper in the late 1980s for “complex” $\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, we solve more involved problems in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of this paper in a much stronger form (even for the initial formulation for distributions of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type). Some results of the present paper are closely related to the famous Beurling–Malliavin theorems on the radius of completeness and a multiplier.

Détail

Keywords: entire function of exponential type, distribution of zeros, subharmonic function of finite type, Riesz distribution of masses
Mots-clés : balayage.