Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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



Model functions with nearly prescribed modulus

Author: Yu. S. Belov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 20 (2008), nomer 2.
Journal: St. Petersburg Math. J. 20 (2009), 163-174
MSC (2000): Primary 30D50, 30D55
Published electronically: January 30, 2009
MathSciNet review: 2423994
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Theta$ be an inner function on the upper half-plane, and let $ K_\Theta = H^2 \ominus \Theta H^2 $ be the corresponding model subspace. A nonnegative measurable function $ \omega$ is said to be strongly admissible for $ K_{\Theta}$ if there exists a nonzero function $ f\in K_{\Theta}$ with $ \vert f\vert\asymp \omega$. Certain conditions sufficient for strong admissibility are given in the case where $ \Theta$ is meromorphic.

References [Enhancements On Off] (What's this?)

  • [BB] A. D. Baranov and A. A. Borichev, Entire functions of exponential type with prescribed modulus on the real axis (unpublished).
  • [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. in J. Math. Sci. (New York). MR 2432176
  • [Bl3] -, Necessary conditions of admissibility for some model subspaces, Algebra i Analiz (to appear).
  • [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)
  • [BBH] A. D. Baranov, A. A. Borichev, and V. P. Havin, Majorants of meromorphic functions with fixed poles, Indiana Univ. Math. J. 56 (2007), 1595-1628. MR 2354693 (2008i:30034)
  • [Cima] J. A. Cima and W. T. Ross, The backward shift on the Hardy space, Math. Surveys Monogr., vol. 79, Amer. Math. Soc., Providence, RI, 2000. MR 1761913 (2002f:47068)
  • [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)
  • [dB] L. de Branges, Hilbert spaces of entire functions, Prentice-Hall, Englewood Cliffs, NJ, 1968. MR 0229011 (37:4590)
  • [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), 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), 1264-1301. MR 2016247 (2004i:30029b)
  • [LS] Yu. I. Lyubarskii and K. Seip, Weighted Paley-Wiener spaces, J. Amer. Math. Soc. 15 (2002), no. 4, 979-1006. MR 1915824 (2003m:46039)
  • [MNH] J. Mashreghi, F. L. Nazarov, and V. P. Havin, 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. Spectral function theory, ``Nauka'', Moscow, 1980; English transl., Grundlehren Math. Wiss., Bd. 273, Springer-Verlag, Berlin, 1986. MR 0575166 (82i:47013); MR 0827223 (87i:47042)
  • [NF] B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, Amer. Elsevier Publ. Co., Inc., New York; Akad. Kiadó, Budapest, 1970. MR 0275190 (43:947)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 30D50, 30D55

Retrieve articles in all journals with MSC (2000): 30D50, 30D55

Additional Information

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

Keywords: Admissible function, Beurling--Malliavin theorem, model subspace, logarithmic integral
Received by editor(s): December 20, 2007
Published electronically: January 30, 2009
Additional Notes: Supported by RFBR, grant no. 06-01-00313
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society