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St. Petersburg Mathematical Journal

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



Admissibility of majorants in certain model subspaces: Necessary conditions

Author: Yu. S. Belov
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 507-525
MSC (2000): Primary 30D50, 30D55, 30D20
Published electronically: June 1, 2009
MathSciNet review: 2473742
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Abstract | References | Similar Articles | Additional Information

Abstract: A nonnegative function $ \omega$ on $ \mathbb{R}$ is called an admissible majorant for an inner function $ \Theta$ if there is a nonzero function $ f\in H^2\ominus \Theta H^2$ such that $ \vert f\vert\le \omega$. Some conditions necessary for admissibility are presented in the case where $ \Theta$ is meromorphic.

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  • [B] A. D. Baranov, Polynomials in the de Branges spaces of entire functions, Ark. Mat. 44 (2006), no. 1, 16-38. MR 2237209 (2007m:46036)
  • [Ba] N. K. Bari, Trigonometric series, Fizmatgiz, Moscow, 1961; English transl., A treatise on trigonometric series. Vols. I, II, The Macmillan Co., New York, 1964. MR 0126115 (23:A3411); MR 0171116 (30:1347)
  • [BB] A. D. Baranov and A. A. Borichev, Entire functions of exponential type with prescribed modulus on the real axis (unpublished).
  • [BBH] A. D. Baranov, A. A. Borichev, and V. P. Havin, Majorants of meromorphic functions with fixed poles, Indiana Univ. Math. J. 56 (2007), no. 4, 1595-1628. MR 2354693 (2008i:30034)
  • [BH] A. D. Baranov and V. P. Khavin, Admissible majorants for model subspaces and arguments of inner functions, Funktsional. Anal. i Prilozhen. 40 (2006), no. 4, 3-21; English transl., Funct. Anal. Appl. 40 (2006), no. 4, 249-263. MR 2307699 (2008c:30058)
  • [Bl1] Yu. S. Belov and V. P. Khavin, On a theorem of I. I. Privalov on the Hilbert transform of Lipschitz functions, Mat. Fiz. Anal. Geom. 11 (2004), no. 4, 380-407. (Russian) MR 2114001 (2005k:26006)
  • [Bl2] Yu. S. Belov, Admissibility criteria for model subspaces with fast growth of the argument of the generating inner function, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 345 (2007), 55-84; English transl., J. Math. Sci. (New York) 148 (2008), no. 6, 813-829. MR 2432176
  • [Bl3] -, Model functions with nearly prescribed modulus, Algebra i Analiz 20 (2008), no. 2, 3-18; English transl., St. Petersburg Math. J. 20 (2009), 163-174. MR 2423994 (2009e:30079)
  • [D] K. M. D'yakonov, Moduli and arguments of analytic functions from subspaces in $ H^p$ that are invariant under the backward shift operator, Sibirsk. Mat. Zh. 31 (1990), no. 6, 64-79; English transl., Siberian Math. J. 31 (1990), no. 6, 926-939 (1991). MR 1097956 (92f:30049)
  • [HJ] V. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, Ergeb. Math. Grenzgeb. (3), Bd. 28, Springer-Verlag, Berlin, 1994. MR 1303780 (96c:42001)
  • [HM1] V. P. Havin and J. Mashreghi, Admissible majorants for model subspaces of $ H^2$. I. Slow winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1231-1263. MR 2016246 (2004i:30029a)
  • [HM2] -, Admissible majorants for model subspaces of $ H^2$. II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1264-1301. MR 2016247 (2004i:30029b)
  • [MNH] J. Mashreghi, F. L. Nazarov, and V. P. Khavin, The Beurling-Malliavin multiplier theorem: The seventh proof, Algebra i Analiz 17 (2005), no. 5, 3-68; English transl., St. Petersburg Math. J. 17 (2006), no. 5, 699-744. MR 2241422 (2007g:42028)
  • [N] N. K. Nikol'skiĭ, Treatise on the shift operator, ``Nauka'', Moscow, 1980; English transl., Grundlehren Math. Wiss., Bd. 273, Springer-Verlag, Berlin, 1986. MR 0575166 (82i:47013); MR 0827223 (87i:47042)

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Additional Information

Yu. S. Belov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 20, Petrodvorets, 198504 St. Petersburg, Russia

Keywords: Blaschke product, model subspace, admissible majorant, Beurling--Malliavin theorem
Received by editor(s): February 20, 2008
Published electronically: June 1, 2009
Additional Notes: Supported by RFBR (grant no. 06–01–00313).
Article copyright: © Copyright 2009 American Mathematical Society

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