A noncommutative extension of Mahler?s interpolation theorem

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 dp 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 dp is replaced by the pro -p metric.