## A convolution equation

Given a rapidly decreasing function f in the euclidean space \R^d, is there a measure \mu solution of the convolution equation exp(-||z||^2/2)*\mu=f?

Here ||z|| stands for the euclidean norm and exp the usual exponential function.

That's basically asking for the Fourier transform of $\mu$ to be a multiple of $\exp(C\|z\|^2) \hat f$ for some $C>0$ (depending on your favorite normalization of the Fourier transform). This imposes a decay requirement on $\hat f$ that is way more stringent than $f$ being rapidly decreasing. – Noam D. Elkies Jan 30 2012 at 3:12