Beurling–Malliavin multiplier theorem: The seventh proof
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J. Mashreghi, F. L. Nazarov and V. P. Havin
Translated by: S. V. Kislyakov - St. Petersburg Math. J. 17 (2006), 699-744
- DOI: https://doi.org/10.1090/S1061-0022-06-00926-5
- Published electronically: July 20, 2006
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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 $H^2(\mathbb {R})$.References
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Bibliographic Information
- J. Mashreghi
- Affiliation: Département de Mathématiques et de Statistique, Université Laval, Laval, Québec G1K7P4, Canada
- MR Author ID: 679575
- Email: Javad.Mashreghi@mat.ulaval.ca
- F. L. Nazarov
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48821
- MR Author ID: 233855
- Email: fedja@math.msu.edu
- V. P. Havin
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198904, Russia
- Email: havin@VH1621.spb.edu
- 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
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 699-744
- MSC (2000): Primary 42A50, 30D55
- DOI: https://doi.org/10.1090/S1061-0022-06-00926-5
- MathSciNet review: 2241422
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya