reshape $(2\pi)^{-n/2} \int_{\mathbb R^n} \mathcal F(\varphi) (\xi) e^{- \frac{\varepsilon^2|\xi|^2}{2}} e^{i\langle x, \xi \rangle} d\xi$

Question

For $\varphi \in \mathcal S(\mathbb R^n)$, $h(x) : = (2\pi)^{-n/2} \int_{\mathbb R^n} \mathcal F(\varphi) (\xi) e^{- \frac{\varepsilon^2|\xi|^2}{2}} e^{i\langle x, \xi \rangle} d\xi$.

Can someone proof me that h(x) = $\varepsilon^{-n}(2\pi)^{-n/2} \int_{\mathbb R^n} \varphi(x+y)e^{- \frac{|y|^2}{2\varepsilon^2}} dy$.

$S(\mathbb R^n)$ the Schwartz space and $|x| := ||x||_2$ the usual multiindex notation. It might be useful that $e^{\frac{-|x|^2}{2}}, x \in \mathbb R^n$ is a eigenfunction of $\mathcal F$ to the eigenvalue 1, but I don't know how to continue with that.

Answer 1

Hint: use the mutipication theore, that says that $$\int_{\mathbb R^n}\mathcal{F}(f)\cdot g=\int_{\mathbb R^n}f\cdot\mathcal{F}(g).$$