A noncommutative extension of Mahler’s theorem on interpolation series

In this paper, we prove an extension of Mahler’s theorem on interpolation series, a celebrated result of pp-adic analysis. Mahler’s original result states that a function from NN to ZZ is uniformly continuous for the pp-adic metric dpdp if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from a free monoid A∗A∗ to ZZ, where dpdp is replaced by the pro-pp metric, the profinite metric on A∗A∗ defined by pp-groups.