Universality properties of the quaternionic power series and entire functions

Abstract

In this paper we establish the existence of “almost universal” quaternionic power series and entire functions. Denoting by B(0, 1) the open unit ball in , this means that there exists a quaternionic power series with radius of convergence 1 such that, denoting by the n‐th partial sum of S, for every , for every axially symmetric open subset Ω of containing K and every f slice regular on Ω, there exists a subsequence of the partial sums of S such that uniformly on K, as . The symbol denotes the set of axially symmetric compact sets in such that is connected for some . This is a slightly weaker property than the classical universal power series phenomenon obtained for analytic only on the interior of K and continuous on K. We also generalize a result originally proven by Birkhoff and finally we show that there exists an entire quaternionic function whose set of derivatives is dense in the class of entire quaternionic functions.