Zeros of Functions in Weighted Spaces with Mixed Norm

In the spaces of analytic functions $f$ in the unit disk with mixed norm and measure satisfying the $\Delta_2$-condition, sharp necessary conditions on subsequences of zeros $\{z_{n_k}(f)\}$ of the function $f$ are obtained in terms of subsequences of numbers $\{n_k\}$. These conditions can be used to define, in the spaces with mixed norm, subsets of functions with certain extremal properties; these subsets provide answers to a number of questions about the zero sets of the spaces under consideration and, in particular, about weighted Bergman spaces.