From answers to a previous question, a finite degree polynomial in the distribution $i\Delta^+(x)$, with Fourier transform $2\pi\delta(k^2-m^2)\theta(k_0)$, is a distribution, even though a product of distributions is not in general well-defined. So far I've not been able to find, or be myself sure enough, whether $\exp{\!\left(i\Delta^+(x)\right)}$ is a distribution. It seems that the exponential is the boundary value of the function $\exp{\!\left(\frac{1}{z^2-m^2}\right)}$, but it's not clear to me yet whether that's enough. There seems to be nothing on this in Streater & Wightman (which I own, but don't yet understand well enough to apply the methods), and I don't have heavy lifting books like Reed & Simon, etc.
More specifically, what I currently think I'm really interested in is whether a bounded function of a distribution, such as $\tan{\!{}^{-1}\left(\exp{\left(i\Delta(x)\right)}\right)}$, is a distribution.