How to prove fourier spectrum symmetrical at k=0

Question

There is this question, Let f(x) be a real function whose Fourier transform is F(k). Prove that the Fourier spectrum (i.e., the graph of |F(k)|^2 versus k) is symmetric around k = 0.

How do I prove it???

Answer 1

I don't think the statement is true. Take any $v\in \mathcal{S}$ (Schwartz space) with support contained in $\mathbb{R}_+$. Since $\mathcal{F}:\mathcal{S}\to\mathcal{S}$, there is $u\in\mathcal{S}$ such that $v=\mathcal{F}u$. Still, $|v|^2$ is not an even function...