Some results concerning the boundary zero sets of general analytic functions

Abstract

Two results concerning the boundary zero sets of analytic functions on the unit disk Δ \Delta are proved. First we consider nonconstant analytic functions f f on Δ \Delta for which the radial limit function f ∗ {f^{\ast }} is defined at each point of the unit circumference C C . We show that a subset E E of C C is the zero set of f ∗ {f^{\ast }} for some such function f f if and only if it is a G δ {\mathcal {G}_\delta } that is not metrically dense in any open arc of C C . We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.