What is the Fourier Transform of $f(x) = x^2$?

Question

What is the Fourier transform of $f(x) = x^2$ ?

I am unsure how the problem is tackled.

Answer 1

Recall that for the function $f_0(x) := x^0 = 1$, we have $$ \def\F{\mathscr F}\F(f_0) = \sqrt{2\pi} \delta $$ Moreover, if $f_1(x) = x$, then for each $u \in \mathscr S'(\mathbf R)$, we have $$ \F(f_1 u) = i\F(u)' $$ Hence, for $f = f_1^2f_0$, we have $$ \F(f_1^2f_0) = i^2\sqrt{2\pi}\delta'' = -\sqrt{2\pi} \delta''$$