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This paper is inspired by a class of infinite order differential operatorsarising in the time evolution of superoscillations. Recently, infinite orderdifferential operators have been considered and characterized on the spaces ofentire monogenic functions, i.e., functions that are in the kernel of the Diracoperators. The focus of this paper is the characterization of infinite orderdifferential operators that act continuously on a different class ofhyperholomorphic functions, called slice hyperholomorphic functions with valuesin a Clifford algebra. We introduce the concept of proximate order andestablish some fundamental properties of entire hyperholomorphic functions thatare crucial for this characterization.

Characterization of continuous homomorphisms on entire slice monogenic functions