Interpolation by sums of series of exponentials and global cauchy problem for convolution operators

TL;DR

This paper investigates the conditions for solving interpolation problems using sums of exponential series and addresses the global Cauchy problem for convolution operators, providing new criteria and strengthening existing results.

Contribution

It introduces new criteria for the solvability of interpolation problems with exponential series and extends the understanding of the global Cauchy problem for convolution operators.

Findings

Derived conditions for interpolation by exponential series on arbitrary node sets.

Established criteria for the solvability of the global Cauchy problem in this context.

Strengthened previous results on convolution operator problems.

Abstract

The study is made of the problem of multiple interpolation on an infinite nodes set by the sums of absolutely convergent series of exponentials whose exponents are from a given set. For entire function conditions on nodes and exponents are obtained that give solubility of the problem. A new approach is demonstrated that enable us, for the case of holomorphic function in a domain, to obtain criteria of solubility of the problem for some class of exponents set and for a special class of nodes set. Upon that the necessity of the conditions is proved in great generality namely for arbitrary nodes sets and in the setting of interpolation by functions that are represented as the Laplace transforms of the Radon measures over the exponents set. Solubility is obtained of the global Cauchy problem for convolution operator with data on the nodes set in domain, in the form of the series of exponentials whose exponents belong to a sparse subset of zero set of characteristic function of the operator. The results substantially strengthen the known results on the theme.