Continuability of multiple power series into sectorial domain by means of interpolation of coefficients

Abstract

We consider the problem on continuability of a multiple power series centered at the origin of $\mathbb{C}^n$ into a sectorial domain. The condition of the mentioned continuability is given in terms of an entire function interpolating the coefficients of the power series. We estimate the indicator function of the interpolating function with the help of which the sectorial set is determined. More precisely, the growth of the interpolating function on the imaginary subspace describes the sectorial set on which the series sum is continued. In the study we use methods of multivariate complex analysis, in particular, integral representations (Cauchy, Mellin, and Lindelof representations), multidimensional residues and properties of power series.