A distance formula for algebras on the disk

Christopher J. Bishop

Vol. 174 (1996), No. 1, 1–27

DOI: 10.2140/pjm.1996.174.1

Abstract

Suppose H^∞(𝔻)[ℱ] is the closed algebra on the disk generated by H^∞(𝔻) and a countable collection ℱ of bounded harmonic functions. Given g ∈ L^∞(𝔻) we give a method for calculating the distance from g to H^∞(𝔻)[ℱ] (in the L^∞ norm). As applications we compute the Bourgain closures of such algebras and give a new proof of a result of Axler and Shields.

Mathematical Subject Classification 2000

Primary: 46J15

Milestones

Received: 23 November 1993

Revised: 14 June 1995

Published: 1 May 1996