A noncommutative extension of Mahler’s interpolation theorem

Abstract

We prove a noncommutative generalization of Mahler’s theorem on interpolation series, a celebrated result of $p$-adic analysis. Mahler’s original result states that a function from $\mathbb{N}$ to $\mathbb{Z}$ is uniformly continuous for the $p$-adic metric $d_p$ if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid A to a free group $F(B)$ where $d_p$ is replaced by the pro-$p$ metric.