Interpolation by sums of the series of exponentials in $H (\mathbb C)$ with interpolation nodes on the rays

TL;DR

This paper addresses the problem of interpolating entire functions using sums of exponential series with nodes on rays, providing solvability criteria and connections to simpler interpolation and approximation problems.

Contribution

It introduces new criteria for the solvability of exponential series interpolation with nodes on rays and links it to simpler interpolation and approximation problems.

Findings

Established a criterion for interpolation solvability on rays

Linked the interpolation problem to simpler point-wise approximation

Provided conditions for the existence of exponential series solutions

Abstract

In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the problem in the case when all infinite portions of interpolation nodes are situated on a finite system of rays. It is also disclosed that the problem is equivalent to particular problems of simple interpolation, as well as of point-wise crude approximation, by sums of the series of exponentials.