The space of arrival for Fourier transform.

Question

If $f\in L^2[-\pi,\pi]$, let $\hat f$ be the Fourier transform of $f$ $$\hat f=\frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ixt} dt, \ \ (-\infty<x<\infty)$$ we can see Fourier transform as an application by $L^2[-\pi,\pi]$ to another space. Usually, who is this space? I imagine that, under certain conditions, it may be (for example) the Paley-Wiener space. Who can tell me something more?