Let $N\in\mathbb N$, let $M\subset\mathbb C^N$ be a $C^\infty$ embedded generic CR submanifold, and let $p\in M$. Let $d:=\operatorname{codim}_{\mathbb R}M$, let $m:=N-d$, and assume $0\le d\le N$, so that $\dim_{\mathbb R}M=2m+d$. Let $T^{0,1}M\subset\mathbb C TM$ be the antiholomorphic CR bundle induced by the embedding, and let $\mu_M:=\mathcal H^{2m+d}|_M$. A function $f\in L^2_{\mathrm{loc}}(U,\mu_M;\mathbb C)$ on an [open set](/page/Open%20Set) $U\subset M$ is called CR in the distribution sense on $U$ if, for every relatively compact open set $O\Subset U$, every local smooth section $\bar L$ of $T^{0,1}M$ defined on a neighbourhood of $\overline O$, and every [test function](/page/Test%20Function) $\psi\in C_c^\infty(O;\mathbb C)$, the [distributional derivative](/page/Distributional%20Derivative) of $f$ in the direction $\bar L$ vanishes on $\psi$ with respect to the smooth density $\mu_M|_O$. Then there exists an open neighbourhood $U\subset M$ of $p$ with the following property: if $f\in L^2_{\mathrm{loc}}(U,\mu_M;\mathbb C)$ is CR in the distribution sense on $U$, then for every open set $V\Subset U$ there is a sequence of holomorphic polynomials $P_k:\mathbb C^N\to\mathbb C$ such that \begin{align*} \lim_{k\to\infty}\|P_k|_M-f\|_{L^2(V,\mu_M)}=0. \end{align*}