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On $ L^1$-norms of meromorphic functions with fixed poles

Author: A. D. Baranov
Journal: Proc. Amer. Math. Soc. 134 (2006), 3003-3013
MSC (2000): Primary 30D50, 30D55; Secondary 46E15, 47B38
Published electronically: May 9, 2006
MathSciNet review: 2231626
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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

Keywords: Blaschke products, shift-coinvariant subspaces, Bernstein's inequality, unconditional basis
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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