  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 $$(a)
\quad|f|\leq \omega\text{\quad or\quad(b)}\quad |f|\asymp\omega.$$ 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 $E$.
