Approximation in weighted Hardy spaces |
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Authors: | A. Bonilla F. Pérez-González A. Stray R. Trujillo-González |
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Affiliation: | (1) Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain;(2) Department of Mathematics, University of Bergen, N-5007 Bergen, Norway |
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Abstract: | This paper is concerned with several approximation problems in the weighted Hardy spacesH p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH p(Ω) coincides, moduloH p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday. |
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