Generic covering properties for spaces of analytic functions

David A. Stegenga and Kenneth R. Stephenson

Vol. 119 (1985), No. 1, 227–243

DOI: 10.2140/pjm.1985.119.227

Abstract

By a classical result of Fatou, a bounded analytic function on the unit disc D, i.e. in the space H^∞(D), has a radial limit at almost every point on ∂D. We examine the question of whether this limiting or boundary value lies in the interior or on the boundary of the image domain. We show that the first case is “typical” in the sense that every function in a certain dense G_δ-set of H^∞ has this property at a.e. boundary point. Several other spaces including the disc algebra and the Dirichlet space are also studied.

Mathematical Subject Classification 2000

Primary: 30D40

Secondary: 30H05

Milestones

Received: 23 November 1983

Revised: 7 February 1984

Published: 1 September 1985