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
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 logω ∈ L 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.
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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 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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Belov, Y., Havin, V. (2014). The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_2-1
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DOI: https://doi.org/10.1007/978-3-0348-0692-3_2-1
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