Let p be a fixed
prime, and let Cp denote the p-adic completion of the algebraic closure of Qp.
For d a fixed positive integer prime to p, set X = Xd=lim←N−Z∕dpNZ.
For example, X1= Zp. We shall first discuss the “inverse Mellin” integral
transform fμ(ρ) =∫Xρ(x)dμ(x) for ρ a Cp-valued bounded measure on
X. We then discuss a second type of p-adic integral transform, which to a
continuous function f(x) on X associates the analytic function whose Taylor
expansion coefficients are f(n). Thirdly, for σ a compact subset of Cp the p-adic
Stieltjes transform φ(z) =∫σ(z − x)−1dμ(x) was shown by Barsky and Vishik
to give a correspondence between measures μ on σ and a certain class of
analytic functions φ on the complement of σ. We shall show that when σ is a
compact subgroup of Cp, the Stieltjes transform is closely related to the first
two transforms. Some examples and arithmetic applications will also be
discussed.
