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

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

Chapter 24 in Operator Theory, 2015, pp 581-607 from Springer

Abstract: Abstract 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.

Keywords: Entire Function; Phase Function; Toeplitz Operator; Blaschke Product; Wiener Space (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_2

Ordering information: This item can be ordered from
http://www.springer.com/9783034806671

DOI: 10.1007/978-3-0348-0667-1_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().