>Now that I read up on it... Actually no. Every tempered distribution has a >Fourier transform, and if that's compactly supported, the original distribution >can be reconstructed via the usual Shannon-Whittaker sinc interpolation >formula. That also goes for polynomials and sine modulated polynomials >in the continuous domain. Whatever that means in general.
Right, that makes intuitive sense. I guess what we lose is the model of sampling as multiplication by a stream of delta functions, but that is more of a pedagogical convenience than a basic requirement to begin with. But what does the convergence of the Shannon-Whittaker formula look like in the case of stuff like polynomials? In the usual setting we get nice results about uniform local convergence, but that requires the asymptotic behavior of the signal being sampled to behave nicely. In a case where it's blowing up at polynomial rate, it seems intuitively that there could be quite strong dependencies on samples far removed in time from any particular region. So the concern would be that it works fine for the ideal sinc interpolator, but could fall apart quite badly for realizable approximations to that. E On Fri, Jun 19, 2015 at 12:49 PM, Sampo Syreeni <de...@iki.fi> wrote: > On 2015-06-12, Ethan Duni wrote: > > Thanks for expanding on that, this is quite interesting stuff. However, >> if I'm following this correctly, it seems to me that the problem of >> multiplication of distributions means that the whole basic set-up of the >> sampling theorem needs to be reworked to make sense in this context. >> > > Now that I read up on it... Actually no. Every tempered distribution has a > Fourier transform, and if that's compactly supported, the original > distribution can be reconstructed via the usual Shannon-Whittaker sinc > interpolation formula. That also goes for polynomials and sine modulated > polynomials in the continuous domain. Whatever that means in general. > > No? >> > > Yes. While the formalism apparently goes through, I don't have the > slightest idea of how to interpret that wrt the usual L^2 theory. I can > sort of get that the polynomial-to-series-of-delta-derivatives duality > works as it should, and via the Schwartz Representation Theorem captures > the asymptotic growth of tempered distributions. But how you'd utilize that > in DSP or with its oversampling problems is thus far beyond me. > > -- > Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front > +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 > -- > 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 > -- 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
