© Springer Basel 2015. All rights are reserved. Let ω be a non-negative function on R. Is it true that there exists a non-zero f from a given space of entire functions X satisfying (a) |f|≤ω or (b) |f| = ω? The classical Beurling-MalliavinMultiplier Theorem corresponds to (a) and the classical Paley-Wiener space as X. This is a survey of recent results for the case when X is a de Branges space H.E/. Numerous answers mainly depend on the behavior of the phase function of the generating function E. For example, if argE is regular, then for any even positive ! non-increasing on [0;∞) with log! Ε L1((1 + x2/-1dx) there exists a non-zero f Ε H(E) such that |f| ≤ |E|ω. This is no longer true for the irregular case. The Toeplitz kernel approach to these problems is discussed. This method was recently developed by N. Makarov and A. Poltoratski.