Constructing continuous functions holomorphic to a curve
type: Item
instance of: scholarly article
title: Constructing continuous functions holomorphic to a curve
zbMATH Open document ID: 0677.46038
DOI: 10.1016/0022-1236(89)90094-3
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: The author presents new interesting proofs of a theorem of \textit{A. Browder} and \textit{J. Wermer} [J. Math. Mech. 12, 119-130 (1963; Zbl 0113.314)], a theorem of A. Davie and a theorem of \textit{T. Gamelin} and \textit{J. Garnett} [J. Funct. Anal. 8, 360-404 (1971; Zbl 0223.30056)] on the characterization of those compact sets \(K\subset \hat C\) for which the algebras \(A_ K\) of all bounded analytic functions on \(\hat C\setminus K\) which have a continuous extension to the closure of \(\hat C\setminus K\) are Dirichlet algebras. The author's methods consist in replacing the use of functional analysis in the original proofs by the explicit construction of bounded solutions of certain \({\bar \partial}\)-problems. They are of a geometrical nature and are intimately related to recent developments in the theory of \(H^{\infty}\) and BMO; see e.g., \textit{J. B. Garnett} and \textit{P. W. Jones} [Acta Math. 155, 27-40 (1985; Zbl 0578.30043)].
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46J30
zbMATH DE Number: 4109382
zbMATH Keywords: Dirichlet algebras
zbMATH Keywords: bounded solutions of certain \({\bar \partial }\)- problems
author: Christopher J. Bishop
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90094-3
OpenAlex ID: W1986816535
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description: scientific article; zbMATH DE number 4109382