Distance Formulas on Weighted Banach Spaces of Analytic Functions
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.
KeywordsBanach spaces of analytic functions Weight Distance Bloch functions
Mathematics Subject ClassificationPrimary 46E15 Secondary 30D45
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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