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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Beurling-Malliavin multiplier theorem: The seventh proof

Authors: J. Mashreghi, F. L. Nazarov and V. P. Havin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 699-744
MSC (2000): Primary 42A50, 30D55
Published electronically: July 20, 2006
MathSciNet review: 2241422
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Abstract: We present a new proof of the Beurling-Malliavin theorem, often called the ``multiplier theorem'', concerning the existence of a real-valued function on $ \mathbb{R}$ with spectrum in a given (small) interval and with a given small majorant of the modulus. This proof pertains entirely to real analysis. It only involves elementary facts about the Hilbert transformation; neither complex variable methods nor potential theory is exploited. The heart of the proof is Theorem 2, which treats preservation of the Lipschitz property under the Hilbert transformation. We also include a short survey of earlier proofs of the Beurling--Malliavin theorem and its generalizations to model (coinvariant) subspaces of the Hardy space $ H^2(\mathbb{R})$.

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Additional Information

J. Mashreghi
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Laval, Québec G1K7P4, Canada

F. L. Nazarov
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48821

V. P. Havin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198904, Russia

Keywords: Fourier transform, spectrum, Hardy space, Paley--Wiener space, Hilbert transform, inner function, outer function, logarithmic integral, Beurling--Malliavin theorem.
Received by editor(s): March 20, 2005
Published electronically: July 20, 2006
Additional Notes: This work was supported by RFBR (grant no. 01-01-00377) and by “Scientific Schools” grant no. Sh-2266.2003.1
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society