# A noncommutative extension of Mahler’s interpolation theorem

### Jean-Éric Pin

Université de Paris et CNRS, France### Christophe Reutenauer

Université du Québec à Montréal, Canada

## 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 $N$ to $Z$ 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.

## Cite this article

Jean-Éric Pin, Christophe Reutenauer, A noncommutative extension of Mahler’s interpolation theorem. J. Noncommut. Geom. 16 (2022), no. 3, pp. 1055–1101

DOI 10.4171/JNCG/480