Locally Analytic Functions over Completions of [formula omitted]r[U

Abstract

It is well known that a continuous functionf:Zp→Qpcan be expanded by Mahler's basis: [formula] withan→0. Amice (Bull. Soc. Math. France92(1964), 117–180) has established conditions on the coefficientsanfor the functionfto be locally analytic, as well as more general results whenZpis replaced by some compact subset of a local field. We study the function field case in this paper. The function field analogue of Mahler's basis is the Carlitz polynomials, and the corre- sponding result for continuous functions has already been established by Wagner (Acta Arith.17(1971), 389–406). We show that the conditions for a continuous function to be locally analytic in the function field case are completely similar to theQpcase. An application to using integral calculus to analytically continue characteristicp-valuedL-series is briefly mentioned at the end of the paper.