Laguerre Entire Functions and Related Locally Convex Spaces

TL;DR

This paper studies Laguerre entire functions within certain Frechet spaces, exploring their properties, the operators acting on them, and applications to solving initial value problems in complex analysis.

Contribution

It introduces a class of infinite order differential operators that preserve Laguerre entire functions and analyzes their role in solving initial value problems.

Findings

Laguerre entire functions form a special subset of exponential type entire functions.

A class of differential operators preserving these functions is characterized.

Applications to initial value problems are demonstrated.

Abstract

A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on compact subsets of the complex plane of polynomials with real nonpositive zeros only, are described. A class of infinite order differential operators which act between these Frechet spaces is inrtoduced and studied. In particular, it is shown that this class preserves the set of Laguerre entire functions. The results obtained are then used to obtain and to study the solutions of a certain initial value problem.