[paurea]

I attacked the same problem somewhat differently and found some interesting groups. You can switch the exponential map you use, use instead a Mittag-Leffler function and find a whole family of groups going from sum to multiplication. Being groups you can define generalized Fourier/Mellin transforms with them. I wrote a blog post explaining the approach (there is a link to the paper at the end for more details). https://paureahack.blogspot.com/2016/06/supersum-subproduct....