[nextaccountic]

> but are now allowed to work with exponentially increasing functions, which standard Fourier theory (even the tempered distributions version) can’t accomodate.

So you can't take the fourier transform of an exponential? But.. it seems you can? https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_...

[enriquto]

exp(-|x|) is not an exponential, it's just the easy, bounded, integrable, half of it :)