Throughout ℤp and ℚp will, respectively, denote the ring of p-adic integers and the field of p-adic numbers (for p prime). We denote by [Copf ]p the completion of the algebraic closure of ℚp with respect to the p-adic metric. Let vp denote the p-adic valuation of [Copf ]p normalised so that vp(p)=1. Put [ ]p ={ω∈[Copf ]p[mid ] ωpn=1 for some n[ges ]0} so that [ ]p is the union of cyclic (multiplicative) groups Cpn of order pn (for n[ges ]0).

Let UD(ℤp) denote the [Copf ]p-algebra of all uniformly differentiable functions f[ratio ]ℤp→[Copf ]p under pointwise addition and convolution multiplication *, where for f, g∈UD(ℤp) and z∈ℤp we have

formula here

the summation being restricted to i, j with vp(i+j−z)[ges ]n.

This situation is a starting point for p-adic Fourier analysis on ℤp, the analogy with the classical (complex) theory being substantially complicated by the absence of a p-adic valued Haar measure on ℤp (see [5, 6] for further details).