Fourier transform of Fourier transform of $f \in L^1\cap L^2$

Question

Let $\mathcal F$ denote the Fourier transfrom, i.e., $\mathcal Ff=\hat f$.

Suppose $f \in L^1 \cap L^2$. By the Plancherel Theorem, $\mathcal Ff\in L^2$ and $\mathcal F \mathcal Ff \in L^2$.

My question is:

$f= \mathcal F \overline{\mathcal Ff}$ a.e. or in $L^2$?

Or at least, $\mathcal F \mathcal Ff \in L^1$?

Thanks and regards

Answer 1

Define $\mathcal F^*f(t)=\mathcal Ff(-t)$. Then $\mathcal F^*\mathcal Ff=f $ in $L^2$ by the Plancherel Theorem.

Hence, $\mathcal F^*\mathcal Ff=f$ a.e. and $\mathcal F^*\mathcal Ff\in L^1\cap L^2$.