  Let $k$ be any number field and $k_{\infty}/k$ any $\mathbb{Z}_p$-extension.
We construct a natural $\Lambda= \mathbb{Z}_p[[ T-1 ]]$-morphism from
$\varprojlim k_n^{\times} \otimes_{\mathbb{Z}} \mathbb{Z}_p$ into a special
subset of $C^1(\mathbb{Z}_p)^*$, the collection of linear functionals on the
set of continuously differentiable functions from $\mathbb{Z}_p \to
\mathbb{C}_p$. We apply the results to the problem of interpolating Gauss sums
attached to Dirichlet characters and the explicit annihilation of real ideal
classes.
