Lacunary series and pseudocontinuations

The main aim of this paper is to prove the following assertion. Let $f=\sum_{n\in E}a_nz^n$ be a function holomorphic and of bounded characteristic in the unit disk $\mathbb D$ where $E$ is a $\Lambda(1)$-subset of $\mathbb Z_+$. Suppose $f$ has a pseudocontinuation of bounded characteristic in an annulus $\{z\in\mathbb C\colon1<|z|<R\}$. Then $f$ admits analytic continuation to the disk $R\mathbb D$. In particular, $f$ is a polynomial if $R=+\infty$. Bibl. 16 titles.