The van der Put base for $C^n$-functions

Abstract

Here n− is defined as follows. For every n ∈ N0, we have a Hensel expansion n = n0 +n1p+ . . .+nsp with ns 6= 0. Then n− = n0 + n1p + . . . + ns−1p s−1. We further put γ0 = 1, γn = n− n− = nsp, δ0 = 1, δn = p s and n∼ = n − δn. Remark that |δn| = |γn|. Let f : Zp → K. The (first) difference quotient φ1f : ∇Zp → K is defined by φ1f(x, y) = f(y)−f(x) y−x , where ∇Zp = Zp × Zp\{(x, x) | x ∈ Zp}. f is called continuously differentiable (or strictly