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-solution 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 $\Delta$ in four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $\Delta = \mathcal {\overline {D}} \mathcal {D}$ 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.