The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces

Let ω be a non-negative function on ℝ $$\mathbb{R}$$ . 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 ? $$\displaystyle{\mbox{ (a)}\quad \vert f\vert \leq \omega \mbox{ or (b)}\quad \vert f\vert \asymp \omega?}$$ The classical Beurling–Malliavin Multiplier 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 ) $$\mathcal{H}(E)$$ . Numerous answers mainly depend on the behavior of the phase function of the generating function E. For example, if arg E $$\arg E$$ is regular, then for any even positive ω non-increasing on [0, ∞) with log ω ∈ L 1 ( ( 1 + x 2 ) − 1 d x ) $$\log \omega \in L^{1}((1 + x^{2})^{-1}dx)$$ there exists a non-zero f ∈ ℋ ( E ) $$f \in \mathcal{H}(E)$$ 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.