Uniqueness theorems for Korenblum type spaces

Abstract

For a scale of spaces $X$ of functions analytic in the unit disc, including the Korenblum space, and for some natural families $\mathcal{E}$ of uniqueness subsets for $X$, we describe minorants for $(X,\mathcal{E})$, that is non-decreasing functions $M:(0,1)\to (0,\infty )$ such that $f\in X$, $E\in \mathcal{E}$, and $\log |f(z)|\le -M(|z|)$ on $E$ imply $f=0$. We give an application of this result to approximation by simple fractions with restrictions on the coefficients.

23 pages