Let $x\in V$ , where $V$ is a real vector space and $dim$ $V=n$ . The Fourier transform is defined as $$\hat f(y)=\int_{x \in V}f(x)e^{2\pi ixy}dx .$$
1) For $f(x)=e^{-\pi x^2}$ I have $$\hat{e^{-\pi y^2}}=\int_{x \in V}e^{-\pi x^2}e^{2\pi i xy}dx=e^{-\pi y^2}\int_{x\in V}e^{-\pi (x-iy)^2}dx=e^{-\pi y^2} .$$
2)For $a>0$ and $f(ax)$ I have $$\hat{f}(ay)=\int_{x\in V}f(ax)e^{2\pi ixy}dx.$$ Set $z=ax$ , $dz=a^ndx$ then
$$ \hat{f}(ay)= \int_{z\in V}f(z)e^{2\pi i \frac{z}{a
}y}a^{-n}dz=\hat{f}(y/a)a^{-n} . $$
Is it correct ?
Thanks for the help .