I understand that for a Schwartz function $f:\mathbb{R}\to\mathbb{C}$, its Fourier transform is another Schwartz function, while if $f$ was a bounded continuous function, then in general its Fourier transform is a distribution. In particular, one can talk about the support of a distribution.
Question: Consider the bounded function $f(x)=\arctan(x)$. Is its distributional Fourier transform $\widehat{f}$ a compactly supported distribution?