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The formal asymptotic expansion of an oscillatory in- tegral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ? on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f ⊗ g 7→ (f ? g)(x) is oscillatory for every x ∈ M.

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Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

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