Let $U \subset \mathbb{R}^n$ be open, let $u \in \mathcal{D}'(U)$, and let $x_0 \in U$. For $\chi \in C_c^\infty(U)$, regard $\chi u$ as a compactly supported distribution on $\mathbb{R}^n$ by extension by zero, and denote its [Fourier transform](/page/Fourier%20Transform) by $\widehat{\chi u}$.
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Then $x_0 \notin \operatorname{sing\,supp} u$ if and only if there exists $\chi \in C_c^\infty(U)$ such that $\chi = 1$ on an open neighbourhood of $x_0$ and, for every $N \in \mathbb{N}$, there exists a constant $C_N > 0$ satisfying