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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On $L^1$-norms of meromorphic functions with fixed poles
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by A. D. Baranov PDF
Proc. Amer. Math. Soc. 134 (2006), 3003-3013 Request permission

Abstract:

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces ${K_B^1}$ generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with $L^1$-norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition $B’\in L^\infty (\mathbb {R})$ for the boundedness of the differentiation on ${K_B^1}$. Our main tool is a construction of an unconditional basis of rational fractions in ${K_B^1}$.
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Additional Information
  • A. D. Baranov
  • Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetskii pr., St. Petersburg, 198504, Russia
  • Address at time of publication: Laboratoire d’Analyse et Géométrie, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence, France
  • Email: antonbaranov@netscape.net
  • Received by editor(s): June 16, 2004
  • Received by editor(s) in revised form: May 9, 2005
  • Published electronically: May 9, 2006
  • Additional Notes: This work was supported in part by RFBR grant 03-01-00377, by the grant for Leading Scientific Schools NSH-2266.2003.1 and by the European Community’s Human Potential Program, contract HPRN-CT-2000-00116 (Analysis and Operators).
  • Communicated by: Juha M. Heinonen
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3003-3013
  • MSC (2000): Primary 30D50, 30D55; Secondary 46E15, 47B38
  • DOI: https://doi.org/10.1090/S0002-9939-06-08349-3
  • MathSciNet review: 2231626