Approximation and interpolation for some spaces of analytic functions in the unit disc

Arne Stray

Vol. 54 (1974), No. 2, 237–251

DOI: 10.2140/pjm.1974.54.237

Abstract

Let U he a bounded open subset of the complex plane C such that U and C∖U are connected. (If B ⊂ C,B denotes its closure in C.) H^∞(U) is the space of all bounded analytic functions defined on U. Let S ⊂ U be She zero set of a nonzero function in H^∞(U). Necessary and sufficient conditions on S are given for the existence of an open sel 0 ⊃_tU∖(S∖S) such that H^∞(0) and H^∞(U) have the same restrictions to S. If U is the unit disc D = {z : |z| < 1} and S is as above, the following result holds for all the Hardy spaces H^p(D), 0 < p ≤∞: Given g ∈ H^p(D), there is a function fanalytic in C∖(S∖S) such thatf|_D ∈ H^p(D) andf = g on S.

Mathematical Subject Classification 2000

Primary: 30A78

Secondary: 46J15

Milestones

Received: 21 November 1972

Published: 1 October 1974