Given Schwartz Space, I'm interested on proving the following:
- The function $f(x) := P(x)\exp[-\alpha|x-a|^2 ] $ where $P$ is a polynomial in $x_1,\ldots,x_N$, $\Re(\alpha)>0$ , and $a \in \mathbb{R} ^N $. I need to prove that $f$ lies in Schwartz space and I have no idea on how to do it.
- If $ f \in L^2 ( \mathbb{R} ^N ) $ has compact support then its Fourier transform is smooth ?
Thanks !