  Motivated by applications to multiplicity formulas in index theory, we study
a family of distributions $\Theta(m;k)$ associated to a piecewise
quasi-polynomial function $m$. The family is indexed by an integer $k \in
\mathbb{Z}_{>0}$, and admits an asymptotic expansion as $k \rightarrow \infty$,
which generalizes the expansion obtained in the Euler-Maclaurin formula. When
$m$ is the multiplicity function arising from the quantization of a symplectic
manifold, the leading term of the asymptotic expansion is the
Duistermaat-Heckman measure. Our main result is that $m$ is uniquely determined
by a collection of such asymptotic expansions. We also show that the
construction is compatible with pushforwards. As an application, we describe a
simpler proof that formal quantization is functorial with respect to
restrictions to a subgroup.
