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

Abstract

The following analogue of Fabry's theorem is proved. Assume that a function ℱ, analytic in the polydisc, has a sufficiently lacunary Taylor series. If on a subset of the torus, of positive Lebesgue measure, the function ℱ coincides in a certain sense with a function analytic in a sufficiently large subset of, then ℱ is analytic in the polydisc for some r >1. As a consequence one obtains that a nonconstant function, analytic in a ball and having a sufficiently lacunary Taylor series, cannot have angular boundary values equal in modulus to unity or having zero real part on a set of positive measure.