The inverse Fueter mapping theorem in integral form using spherical monogenics

Abstract

In this paper we prove an integral representation formula for the inverse Fueter mapping theorem for monogenic functions defined on axially symmetric open sets U ⊆ ℝⁿ⁺¹, i.e. on open sets U invariant under the action of SO(n), where n is an odd number. Every monogenic function on such an open set U can be written as a series of axially monogenic functions of degree k, i.e. functions of type $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x) = \left[ {A\left( {x_{0,\rho } } \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } {\rm B}\left( {x_{0,\rho } } \right)} \right]\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$, where A(x ₀, ρ) and B(x ₀, ρ) satisfy a suitable Vekua-type system and $\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$ is a homogeneous monogenic polynomial of degree k. The Fueter mapping theorem says that given a holomorphic function f of a paravector variable defined on U, then the function $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$ given by

$$\Delta ^{k + \tfrac{{n - 1}} {2}} \left( {f(x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )} \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$$

is a monogenic function. The aim of this paper is to invert the Fueter mapping theorem by determining a holomorphic function f of a paravector variable in terms of $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{f} _k (x)\mathcal{P}_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )$. This result allows one to invert the Fueter mapping theorem for any monogenic function defined on an axially symmetric open set.