Sobolev approximation by a sum of subalgebras on the circle

John Brady Garnett and Anthony G. O’Farrell

Vol. 65 (1976), No. 1, 55–63

DOI: 10.2140/pjm.1976.65.55

Abstract

Let ψ be an orientation-reversing homeomorphism of the unit circle onto itself. We consider approximation in certain Sobolev norms by functions of the form f(z) + g(ψ), where f and g are polynomials. The methods involve conformal welding and Hardy space theory. We construct a Jordan arc of positive continuous analytic capacity such that the harmonic measures for the two complementary domains are mutually absolutely continuous.

Mathematical Subject Classification

Primary: 30A82, 30A82

Milestones

Received: 14 April 1975

Revised: 18 February 1976

Published: 1 July 1976