  We introduce the notion of an oscillatory formal distribution supported at a
point. We prove that a formal distribution is given by a formal oscillatory
integral if and only if it is an oscillatory distribution that has a certain
nondegeneracy property. We give an algorithm that recovers the jet of infinite
order of the integral kernel of a formal oscillatory integral at the critical
point from the corresponding formal distribution. We also prove that a star
product $\star$ on a Poisson manifold $M$ is natural in the sense of Gutt and
Rawnsley if and only if the formal distribution $f \otimes g \mapsto (f \star
g)(x)$ is oscillatory for every $x \in M$.
