  We consider the basic heat operator on functions on a Riemannian foliation of
a compact, Riemannian manifold, and we show that the trace of this operator has
a particular short time asymptotic expansion. The coefficients in this
expansion are obtainable from local transverse geometric invariants - functions
computable by analyzing the manifold in an arbitrarily small neighborhood of a
leaf closure. Using this expansion, we prove some results about the spectrum of
the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also,
we explicitly calculate the first two nontrivial coefficients of the expansion
for special cases such as codimension two foliations and foliations with
regular closure.
