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

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Operator Theory

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

$$\displaystyle{\mbox{ (a)}\quad \vert f\vert \leq \omega \mbox{ or (b)}\quad \vert f\vert \asymp \omega?}$$

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 \(\mathcal{H}(E)\). 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 \(\log \omega \in L^{1}((1 + x^{2})^{-1}dx)\) 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.

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Acknowledgements

We thank A. Borichev for the permission to expose his construction illustrating the sharpness of the BM-theorem (see section “More on the Oscillations of BM-Majorants: Borichev’s Construction”). The first author was supported by the Chebyshev Laboratory (St. Petersburg State University) under RF Government grant 11.G34.31.0026, by JSC “Gazprom Neft,” and by RFBR grant 12-01-31492. The second author was supported by St. Petersburg State University Action Item 2: NIR “Function theory, operators theory and its applications” 6.38.78.2011.

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Authors and Affiliations

  1. Chebyshev Laboratory, St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia

    Yurii Belov

  2. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Victor Havin

Authors
  1. Yurii Belov
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  2. Victor Havin
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Correspondence to Yurii Belov .

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Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

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Belov, Y., Havin, V. (2015). The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_2

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