Representation of Analytic Functions by Series of Exponential Monomials in Convex Domains and Its Applications

Abstract

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.