Rudin’s orthogonality problem and the Nevanlinna counting function

Author:

Bourdon Paul

Abstract

Let ϕ \phi be a holomorphic function taking the open unit disk U U into itself. We show that the set of nonnegative powers of ϕ \phi is orthogonal in L 2 ( U ) L^2(\partial U) if and only if the Nevanlinna counting function of ϕ \phi , N ϕ N_\phi , is essentially radial. As a corollary, we obtain that the orthogonality of { ϕ n : n = 0 , 1 , 2 , } \{\phi ^n: n=0,1,2,\ldots \} for a univalent ϕ \phi implies ϕ ( z ) = α z \phi (z) = \alpha z for some constant α \alpha . We also show that if { ϕ n : n = 0 , 1 , 2 , } \{\phi ^n: n=0,1,2,\ldots \} is orthogonal, then the closure of ϕ ( U ) \phi (U) must be a disk.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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