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

  • Yurii Belov
  • Victor Havin
Living reference work entry

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.

Keywords

Entire Function Phase Function Toeplitz Operator Blaschke Product Wiener Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

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 B M -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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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Chebyshev Laboratory, St. Petersburg State UniversityVasilyevsky Island, St. PetersburgRussia
  2. 2.Department of Mathematics and Mechanics, St. Petersburg State UniversitySt. PetersburgRussia

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