  A famous theorem of Dixmier-Malliavin asserts that every smooth,
compactly-supported function on a Lie group can be expressed as a finite sum in
which each term is the convolution, with respect to Haar measure, of two such
functions. We establish that the same holds for a Lie groupoid. Most of the
heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We
also need the technology of Lie algebroids and the corresponding notion of
exponential map. As an application, we obtain a result on the arithmetic of
ideals in the smooth convolution algebra of a Lie groupoid arising from
functions vanishing to given order on an invariant submanifold of the unit
space.
