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 $x \in M$.

中文翻译：

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正式的振荡分布

我们引入了一个点支持的振荡形式分布的概念。我们证明了一个正式的分布是由一个正式的振荡积分给出的，当且仅当它是一个具有一定非简并性质的振荡分布。我们给出了一种算法，该算法可以从相应的形式分布中恢复临界点处形式振荡积分的积分核的无限阶射流。我们还证明了泊松流形 $M$ 上的星积 $\star$ 在 Gutt 和 Rawnsley 意义上是自然的当且仅当形式分布 $f \otimes g \mapsto (f \star g)(x) $ 对于每 $x \in M$ 都是震荡的。