Arakelian's approximation theorem of Runge type in the hypercomplex setting

In this paper we prove Arakelian's approximation theorem of Runge type in the framework of slice hyperholomorphic functions, namely for slice regular functions of a quaternionic variable and for slice monogenic functions of a paravector variable. Specifically, we show that any slice hyperholomorphic function in a neighborhood of a so-called Arakelian (closed and possible unbounded) set E, can be uniformly approximated on E by sequences of entire slice hyperholomorphic functions.