Living Reference Work Entry

Operator Theory

pp 1-24

Date: Latest Version

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

  • Yurii BelovAffiliated withChebyshev Laboratory, St. Petersburg State University Email author 
  • , Victor HavinAffiliated withDepartment of Mathematics and Mechanics, St. Petersburg State University

Abstract

Let ω be a non-negative function on \(\mathbb{R}\). Is it true that there exists a non-zero f from a given space of entire functions X satisfying
$$\displaystyle{\text{(a)}\quad \vert f\vert \leq \omega \quad \mathrm{or\quad (b)}\quad \vert f\vert \asymp \omega ?}$$
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 \(\mathcal{H}(E)\). Numerous answers mainly depend on the behavior of the phase function of the generating function E .

For example, if \(\arg E\) is regular, then for any even positive ω non-increasing on [0 , ) with logωL 1((1 + x 2)−1dx) there exists a non-zero \(f \in \mathcal{H}(E)\) 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.