Approximation in weighted Hardy spaces

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 73; no. 1; pp. 65 - 89
• Main Authors: Bonilla, A., Pérez-González, F., Stray, A., Trujillo-González, R.
• Format: Journal Article
• Language: English
• Published: Jerusalem Springer Nature B.V 01.01.1997
• Subjects: Analytic functions; Approximation; Continuity (mathematics); Mathematical analysis; Polynomials; Set theory
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/BF02788138

This paper is concerned with several approximation problems in the weighted Hardy spacesH^sup p^([Omega]) of analytic functions in the open unit disc D of the complex plane . We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH^sup p^([Omega]) coincides, moduloH^sup p^([Omega]), 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^sup p^([Omega]), 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^sup p^(D), 1 p < ∞.[PUBLICATION ABSTRACT]