  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.
