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Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 893–900 | Cite as

Distance Formulas on Weighted Banach Spaces of Analytic Functions

  • José BonetEmail author
  • Wolfgang Lusky
  • Jari Taskinen
Article
  • 91 Downloads

Abstract

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function \(f_0\) in the Banach space \(H_v^\infty \) of all analytic functions f such that v|f| is bounded, the distance of \(f_0\) to the subspace \(H_v^0\) of \(H_v^\infty \) of all the functions g such that v|g| vanishes at infinity is attained at a function \(g_0 \in H_v^0\). Moreover a simple, direct proof of the formula of the distance of f to \(H_v^0\) due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.

Keywords

Banach spaces of analytic functions Weight Distance Bloch functions 

Mathematics Subject Classification

Primary 46E15 Secondary 30D45 

Notes

Acknowledgements

The authors are very thankful to the referees for their careful reading of the manuscript and their suggestions. The research of Bonet was partially supported by the Projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y Aplicada IUMPAUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany
  3. 3.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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