Orthogonal Functions in H

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 μf in the plane, and we characterize which symmetric measures can occur in this way. Such functions have the property that { f n} 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.