A polyanalytic functional calculus of order 2 on the S-spectrum

Abstract

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in ${\mathbb{R}}^4$, denoted by $\mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($\Delta$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $\Delta =\overline{\mathcal{D}}\mathcal{D}$ to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of ${\mathcal{D}}^2$. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the $S$-spectrum. As far as the authors know, this is the first time that a monogenic polyanalytic functional calculus has been taken into consideration.

arXiv admin note: text overlap with arXiv:2205.08162