Orthogonal functions in H^∞

Christopher J. Bishop

Vol. 220 (2005), No. 1, 1–31

DOI: 10.2140/pjm.2005.220.1

Abstract

We construct examples of H^∞ functions f on the unit disk such that the push-forward of Lebesgue measure on the circle is a radially symmetric measure μ_f in the plane, and we characterize which symmetric measures can occur in this way. Such functions have the property that {fⁿ} is orthogonal in H², and provide counterexamples to a conjecture of W. Rudin, independently disproved by Carl Sundberg. Among the consequences is that there is an f in the unit ball of H^∞ such that the corresponding composition operator maps the Bergman space isometrically into a closed subspace of the Hardy space.

Keywords

Rudin’s conjecture, orthogonal functions, cut-and-paste construction, composition operators, radial measures, Nevalinna function, harmonic measure, Bergmann space, Hardy space

Mathematical Subject Classification 2000

Primary: 30H05

Secondary: 30D35, 30D55, 47B38

Milestones

Received: 1 May 2002

Revised: 1 October 2002

Accepted: 29 November 2004

Published: 1 May 2005