TITLE: Exponential derivative of delta distribution?
QUESTION [5 upvotes]: This question is from here. I'm asking it here as well to increase the number of people who see it and might be able to help.
The question is, what is the result of the following integral for integer $n$ and real $x$?
$$\int_{-\infty}^\infty dy\, e^{ny}e^{iyx}$$
Does it diverge and give infinity, or is it actually just equal to the following?
$$=\sum_{m=0}^\infty\frac{n^m}{m!}\int_{-\infty}^\infty dy\, y^m e^{iyx}=2\pi\sum_{m=0}^\infty\frac{1}{m!}\left(-in\frac{\partial}{\partial x}\right)^m\delta(x)=2\pi \left(e^{-in\frac{\partial}{\partial x}}\delta(x)\right)$$
More importantly, is there some book or scientific article on distributions that discusses this? Thanks for any suggestion!

REPLY [3 votes]: I think Schwartz and Gelfand-Shilov, et al, had already considered the possibility of having "Fourier transform" map from all distributions to the dual of the Paley-Wiener space (the latter being the image of test functions under literal Fourier transform).
So although the literal integral you write does not converge (nor do many for tempered distributions, either), it does have a sense compatible with other things.