Some results concerning the boundary zero sets of general analytic functions

Berman, Robert D.

doi:10.1090/S0002-9947-1986-0816329-6

Some results concerning the boundary zero sets of general analytic functions
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Trans. Amer. Math. Soc. 293 (1986), 827-836 Request permission

Abstract:

Two results concerning the boundary zero sets of analytic functions on the unit disk $\Delta$ are proved. First we consider nonconstant analytic functions $f$ on $\Delta$ for which the radial limit function ${f^{\ast }}$ is defined at each point of the unit circumference $C$. We show that a subset $E$ of $C$ is the zero set of ${f^{\ast }}$ for some such function $f$ if and only if it is a ${\mathcal {G}_\delta }$ that is not metrically dense in any open arc of $C$. 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.

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