Entire monogenic functions of given proximate order and continuous homomorphisms

TL;DR

This paper studies the continuity of infinite order differential operators on spaces of monogenic entire functions, introducing the concept of proximate order to characterize such operators in the context of hyperholomorphic functions.

Contribution

It introduces the proximate order for monogenic functions and characterizes infinite order differential operators acting continuously on these function spaces.

Findings

Proximate order is fundamental for analyzing monogenic functions.

Characterization of continuous infinite order differential operators.

Application to hyperholomorphic function spaces.

Abstract

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schr\"odinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As important application we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.