operator core of a fourier multiplier

Question

Let $\Lambda:\mathcal{D}(\Lambda)\to L^2(\mathbb{R})$ be a densely defined Fourier multiplier, i.e, for any $u\in\mathcal{D}(\Lambda)$, we have $\mathcal{F}(\Lambda u)=m_{\Lambda}\mathcal{F}(u)$, where $\mathcal{F}$ denotes the Fourier transform. We take $\mathcal{D}(\Lambda)=\{u\in L^2(\mathbb{R}): m_{\Lambda}\mathcal{F}(u)\in L^2(\mathbb{R})\}$. Let $\varphi(x)=e^{-\frac{x^2}{2}}$. By Wiener's Tauberian theorem, we know that the set $\mathcal{G}=\operatorname{Span}\{\varphi(\cdot+a): a\in\mathbb{R}\}$ is dense in $L^2(\mathbb{R})$. Now, assume that $|m_{\Lambda}(x)|\le Ae^{B|x|}$ for all $x\in\mathbb{R}$ and for some constant $A,B$ and it is non-vanishing almost everywhere. Can we show that $\mathcal{G}$ is a core of the operator $(\Lambda,\mathcal{D}(\Lambda))$?