Infinite-order Differential Operators Acting on Entire Hyperholomorphic Functions

Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger 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. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions.