The Beurling--Malliavin Multiplier Theorem and its analogs for the de Branges spaces

Abstract

Let $\omega$ be a non-negative function on $\mathbb{R}$. We are looking for a non-zero $f$ from a given space of entire functions $X$ satisfying $$(a)|f|\le \omega \text{or(b)}|f|\asymp \omega .$$ The classical Beurling--Malliavin Multiplier Theorem corresponds to $(a)$ and the classical Paley--Wiener space as $X$. We survey recent results for the case when $X$ is a de Branges space $\he$. Numerous answers mainly depend on the behaviour of the phase function of the generating function $E$.

arXiv commentSurvey, 25 pages