Infinite order differential operators acting on entire hyperholomorphic functions

TL;DR

This paper studies the continuity of infinite order differential operators on entire hyperholomorphic functions, extending concepts from complex analysis and quantum mechanics to functions of paravector variables.

Contribution

It investigates the continuity of infinite order differential operators on hyperholomorphic functions, highlighting the role of exponential bounds and monogenic functions in this context.

Findings

Entire hyperholomorphic functions with exponential bounds are crucial for operator continuity.

Exponential functions of paravector variables are not in the Dirac operator's kernel.

Entire monogenic functions with exponential bounds are significant in the theory.

Abstract

Infinite order differential operators appear in different fields of Mathematics and Physics and in the last decades they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schr\"odinger equation. Inspired by the 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. The two classes of hyperholomorphic functions, that constitute a natural extension of functions ofone complex variable to functions of paravector variables are illustrated by the Fueter-Sce-Qian mapping theorem. We show that, even though the two notions of hyperholomorphic functions are quite different from each other, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite order differential operators acting on these two classes of entire hyperholomorphic functions. We point out the remarkable fact that the exponential function of a paravector variable is not in the kernel of the Dirac operator but entire monogenic functions with exponential bounds play an important role in the theory.