One property of zero set of function invertible in the sense of Ehrenpreis in the Schwartz algebra

TL;DR

This paper investigates the properties of zero sets of functions in the Schwartz algebra that are invertible in Ehrenpreis's sense, focusing on zeros near the real axis, and provides new insights into their structure.

Contribution

It establishes specific properties of zero subsets of Ehrenpreis-invertible functions in the Schwartz algebra, advancing understanding of their zero distribution.

Findings

Zeros of invertible functions lie close to the real axis

Zero sets exhibit particular structural properties

Results contribute to the theory of Schwartz algebra functions

Abstract

We consider those elements of the Schwartz algebra of entire functions which are Fourier-Laplace transforms of invertible distributions with compact supports on the real line. These functions are called invertible in the sense of Ehrenpreis. The presented result concerns with the properties of zero subsets of invertible in the sense of Ehrenpreis function f. Namely, we establish some properties of the zero subset formed by zeros of f laying not far from the real axis.