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

Yurii Belov () and Victor Havin ()
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Yurii Belov: St. Petersburg State University, Chebyshev Laboratory
Victor Havin: St. Petersburg State University, Department of Mathematics and Mechanics

Chapter 17 in Operator Theory, 2026, pp 395-421 from Springer

Abstract: Abstract Let ω $$\omega $$ 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 ? $$\displaystyle \text{(a)} \quad |f|\leq \omega \text{\quad or\quad (b)}\quad |f|\asymp \omega ? $$ The classical Beurling–Malliavin Multiplier Theorem corresponds to (a) and the classical Paley–Wiener space as X. 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 $$\omega $$ non-increasing on [ 0 , ∞ ) $$[0,\infty )$$ with log ω ∈ L 1 ( ( 1 + x 2 ) −1 dx ) $$\log \omega \in L^1((1+x^2)^{-1}dx)$$ there exists a non-zero f ∈ ℋ ( E ) $$f\in \mathcal {H}(E)$$ such that | f | ≤ | E | ω $$|f|\leq |E|\omega $$ . 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.

Date: 2026
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DOI: 10.1007/978-3-032-16356-1_2

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