Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$, let $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$, and for a multi-index $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{N}_0^n$ write $|\alpha|=\alpha_1+\cdots+\alpha_n$, $\xi^\alpha=\xi_1^{\alpha_1}\cdots\xi_n^{\alpha_n}$, and $D^\alpha=\partial_{x_1}^{\alpha_1}\cdots\partial_{x_n}^{\alpha_n}$. Let $\mathcal{S}(\mathbb{R}^n)$ denote the [Schwartz space](/page/Schwartz%20Space) of rapidly decreasing smooth functions on $\mathbb{R}^n$, and let $\mathcal{S}'(\mathbb{R}^n)$ denote its tempered-distribution dual. Let $\mathcal{F}:\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ denote the distributional [Fourier transform](/page/Fourier%20Transform) with symmetric normalization, so that for integrable functions $f$ one has $\widehat f(\xi)=(2\pi)^{-n/2}\int_{\mathbb{R}^n} f(x)e^{-ix\cdot\xi}\,d\mathcal{L}^n(x)$, and let $\mathcal{F}^{-1}$ denote its inverse, represented on integrable functions $F$ by $\mathcal{F}^{-1}F(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}F(\xi)e^{ix\cdot\xi}\,d\mathcal{L}^n(\xi)$. Let $u \in \mathcal{D}'(U)$, and let $x_0 \in U$. For $\chi \in C_c^\infty(U)$, define the zero extension of $\chi u$ to $\mathbb{R}^n$ to be the compactly supported distribution $T_\chi \in \mathcal{D}'(\mathbb{R}^n)$ given by $T_\chi(\phi)=u((\chi\phi)|_U)$ for every $\phi \in C_c^\infty(\mathbb{R}^n)$; this compactly supported distribution is viewed as an element of $\mathcal{S}'(\mathbb{R}^n)$. Define $\widehat{\chi u}:=\widehat{T_\chi}$, and say that $\widehat{\chi u}$ is rapidly decreasing when it is represented by a function $F: \mathbb{R}^n \to \mathbb{C}$ such that, for every integer $N \geq 0$, there exists a constant $C_N > 0$ with

\begin{align*} |F(\xi)| \leq C_N(1 + |\xi|)^{-N} \end{align*}

for every $\xi \in \mathbb{R}^n$. 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 $\widehat{\chi u}$ is rapidly decreasing.