The Malliavin-Rubel theorem on small entire functions of exponential type with given zeros: 60 years later

Abstract

Let $Z$ and $W$ are distributions of points on the complex plane $\mathbb C$. The following problem goes back to the studies of F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane and others. For which $Z$ and $W$ for an entire function $g\neq 0$ of exponential type vanishing on $W$, there is an entire function $f\neq 0$ of exponential type vanishing on $Z$ such that $|f|\leq |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $Z$ and $W$ lying only on the positive semiaxis. A number of generalizations of this criterion were established by us in the late 1980s for "complex" $Z\subset\mathbb C$ and $W\subset\mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, tougher problems are solved 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 the paper in a much stronger form even in the initial formulation for distributions of points $Z$ and $W$ and entire functions $f$ and $g$ of exponential type. Some of the results of the article are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and the multiplier.