Invariant subspaces in half-plane

Abstract

In this work we consider sequences of specified order $\rho(r)$. We find necessary and sufficient conditions guaranteeing that a sequence $\Lambda^2\supseteq\Lambda^1$ consists a regularly distributed set $\Lambda$ with a prescribed angular density containing $\Lambda^1$. These results cover a most part of knonw results on constructions of regularly distributed sets. We consider various applications of the results. On the base of them, we prove theorems on splitting of entire functions of a specified order $\rho(r)$. Moreover, we find an asymptotic representation of an entire function with a measurable sequence of zeroes. This generalizes a classical representation by B.Ya. Levin with a regularly distributed zero set to the case of a function with a measurable zero set. This representation is based on the obtained representation for a function, the zero set of which has a zero density. Its implication is the strengthening of a known result by M.L. Cartwright on the type of a function with a zero set having a zero density. Another corollary is the way for constructing entire functions of exponential type with a prescribed indicator and the minimal possible zero density.