# Fourier Transform of $L^2$ functions

Let $\Omega\subset\mathbb{R}^3$ be a bounded regular domain. Take $f(x,t)$ to be a function defined on $\Omega\times (0,T)$. Suppose that $f\in L^2(\Omega)$, and let $\hat{f}$ be the Fourier transform of $f$ in time (being extended by $0$ outside $(0,T)$). Can we prove that $\hat{f}$ remains also in $L^2(\Omega)$?

• Hello and welcome to math.stackexchange. What do you mean by $f \in L^2(\Omega)$? You are given a function that is defined on $\Omega \times (0,T)$. – Hans Engler Jul 16 '17 at 17:49
• Should it be $f (\cdot, t) \in L^2 (\Omega)$ for all $t \in (0, T)$, i.e. that $f$ is $L^2$ in $x$? – md2perpe Jul 26 '17 at 17:15