Fourier Transform of the given generalised function

Question

One of a problem requires the following equation to be true. $$\int\limits_{-\infty}^{\infty}\; f(x)\;e^{\pi\iota x^2}\;e^{-2\pi\iota ux}\;dx = F(u)\; e^{-\pi\iota u^2}$$ where $F(u)$ is the Fourier Transform of $f(x)$ If this is true, can anyone please provide a proof.

Answer 1

This is false. If you take $f(x) = g(x) e^{-i\pi x^2}$, then the left is the Fourier transform of $g$ while the right is $g(x) e^{-2 i\pi x^2}$