On 11 Jun 2015, at 19:58, Sampo Syreeni <de...@iki.fi> wrote: > Now, I don't know whether there is a framework out there which can handle > plain exponentials, a well as tempered distributions handle at most > polynomial growth. I suspect not, because that would call for the test > functions to be faster decaying than any exponential, and such functions are > measure theoretically tricky at best. I suspect what you'd at best arrive is > would seem very much like the L_p theory or the Laplace transform: various > exponential growth rates being quantified by various upper limits of > regularization, and so not one single theory where the Fourier transform > exists for all your functions at the same time, and the whole thing > restricting to the nice L_2 isometry where both functions belong to that > space.
I think it’s not hard to prove that there is no consistent generalisation of the Fourier transform or regularisation method that would allow plain exponentials. Take a look at the representation of the time derivative operator in both time domain, d/dt, and frequency domain, i*omega. The one-dimensional eigensubspaces of i*omega are spanned by the eigenvectors delta(omega-omega0) with the associated eigenvalues i*omega0. That means all eigenvalues are necessarily imaginary, with exception of omega0=0. On the other hand, exp(t) is an eigenvector of d/dt with eigenvalue 1, which is not part of the spectrum of the frequency domain representation. This means, there is no analytic continuation from other transforms, no regularisation or transform in a weaker distributional sense. Hope this helps, Andreas -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
