Pseudocontinuation and properties of analytic functions on boundary sets of positive measure

B. Jöricke

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 88-96

Résumé

The following analog of Fabry's theorem is proved: If a function $\mathcal F$ analytic in the polydise $\mathbb D^n$ has a very lacunary Taylor series and coincides (in a certain sense) on a set of positive measure in $\mathbb T^n$ with a function analytic in a sufficiently large subset of $(\mathbb C\setminus\bar{\mathbb D})^n$ then $\mathcal F$ is analytic in the polydisc $(r\mathbb D)^n$ with $r>1$. This implies that a nonconstant analytic function in the ball $B\subset\mathbb C^n$ with very lacunary Taylor series cannot have nontangential boundary walues with constant modulus or zero real part on a set of positive measure.