Espaces de fonctions de classe Cr sur OF

In this paper we introduce a class of Banach spaces of functions of class Cr (where r is a positive real number) and the associated dual spaces of distributions of order r, which turn out to be useful in p-adic Langlands theory De Ieso (2013) [9]. We construct a Banach basis for these spaces and we give a criterion for telling when a linear form on a space of locally Qp-polynomial functions extends to a distribution of order r. This generalises the classical results of Amice-Vélu (1975) and Vishik (1976) [1,18].