Is Riemann–Lebesgue lemma valuble in $L2(\mathbb{R})$

Question

If $f\in L_1$ on $\mathbb{R}$, that is to say, if the Lebesgue integral of $|f|$ is finite, then the Fourier transform of $f$ satisfies $$\hat{f}(z):= \int_{\mathbb{R}} f(x)e^{-izx} dx \rightarrow 0, \text{when}~\, |z| \rightarrow \infty. $$ This result is known as Riemann–Lebesgue lemma. I was wondering if this result is also true for $f\in L_2$ on $\mathbb{R}$, that is, the Lebesgue integral of $|f|^2$ is finite. Can anybody help me?

Answer 1

Since there is a Fourier inversion formula on $L^2$, it means that every $L^2$ function is a Fourier transform of some $L^2$ function. So if every such Fourier transform converged to zero at infinity, it would mean that every $L^2$ function does too.

But this is obviously not the case without regularity assumptions on the function.