Invertibility In the Sense of Ehrenpreis

TL;DR

This paper investigates the conditions under which certain zero sets of entire functions in the Schwartz algebra correspond to invertible elements in the sense of Ehrenpreis, focusing on the properties of slowly decreasing functions.

Contribution

It provides new criteria for when a complex sequence can be realized as the zero set of an invertible Schwartz algebra element in the Ehrenpreis sense.

Findings

Characterization of zero sets for invertible Schwartz algebra elements

Conditions for sequences to be zero sets of Ehrenpreis invertible functions

Insights into the structure of the Schwartz algebra and its invertible elements

Abstract

We consider zero sets of entire functions belonging to the Schwartz algebra. This algebra is defined as the Fourier-Laplace transform image of the space of all distributions compactly supported on the real line. We study the conditions under which given complex sequence forms zero set of some invertible in the sense of Ehrenpreis element of the Schwartz algebra (slowly decreasing function).