Rudin Orthogonality On The Dirichlet Space

This paper studies the Rudin orthogonality problem on the Dirichlet space:For what functions φ bounded and analytic in the unit disk, does {φn:n= 0,1,2,...} form an orthogonal set in the Dirichlet space? For an analytic self-map ip fixing zero in the Dirichlet space, we obtain a complete characterization of the orthogonality of {φn:n= 0,1,2,...} in the Dirichlet space in terms of its counting function and induced measure respectively. As applications, we show that if φ is a bounded analytic function on the closed unit disk, fixing zero, then {φn:n= 0,1,2,...} is orthogonal in the Dirichlet space if and only if φ is a constant multiple of some finite Blaschke product. We also prove that if φ is an univalent analytic self-map on the unit disk, fixing zero, then {φ:n= 0,1,2,...} is orthogonal in the Dirichlet space if and only if the area measure of Dr\φ(D) is zero, where Dr={z:|z|<||φ||∞}.