Interpolation by Weakly Differentiable Functions on Banach Spaces

Let (an) be a weakly null Schauder basis of a Banach space E, and let (λn) be a convergent sequence of real numbers. We study the problem of finding an m-times weakly uniformly differentiable function ƒ on E such that ƒ(an) = λn. We prove that this problem has always a solution for m = 1. In some cases we find a solution for m = ∞, for instance when E is super-reflexive or when (an) is a symmetric basis and E does not contain a copy of c0. In these cases we obtain as a consequence the nonreflexivity of the space of infinitely weakly uniformly differentiable functions on E.