>The fact that the constant maps to a delta and the successive higher >derivatives to monomials of equally higher order sort of correspond to >the fact that in order to approximate something with such fiendishly >local structure as a delta (corresponding in convolution to taking the >value) and its derivatives (convolution with which is the derivative of >corresponding order) calls for polynomially increasing amounts of high >frequency energy. That is, something you can only handle in the distributional >setting, with its functionals and only a local sense of integration. Trying to >interpret something like that the way we do in conventional L_2 theory sounds >ikely to lead to pain.
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. I.e., not much point worrying about whether to call whatever exotic combination of derivatives of delta functions that result from polynomials as "band limited" or not, when we don't have a way to relate such a property back to sampling/reconstruction of well-tempered distributions in the first place. No? E On Thu, Jun 11, 2015 at 2:00 AM, Sampo Syreeni <de...@iki.fi> wrote: > On 2015-06-09, Ethan Duni wrote: > > The Fourier transform does not exist for functions that blow up to +- >> infinity like that. To do frequency domain analysis of those kinds of >> signals, you need to use the Laplace and/or Z transforms. >> > > Actually in the distributional setting polynomials do have Fourier > transforms. Naked exponentials, no, but those are evil to begin with. The > reason that works is the duality between the Schwartz space and that of > tempered distributions themselves. The test functions are required to be > rapidly decreasing which means that integrals between them and any function > of at most polynomial growth converge, and so polynomials induce perfectly > well behaved distributions. In essence the regularization which the Laplace > transform gets from its exponential term and varying area of convergence is > taken care of by the structure of the Schwartz space, and the whole > machinery implements not a global theory of integration, but a local one. > > I don't know how useful the resulting Fourier transforms would be to the > original poster, though: their structure is weird to say the least. Under > the Fourier transform polynomials map to linear combinations of the > derivatives of various orders of the delta distribution, and their spectrum > has as its support the single point x=0. The same goes the other way: > derivatives map to monomials of corresponding order. In a vague sense that > functional structure at a certain frequency corresponds to the asymptotic > behavior of the distribution, while the tamer function-like part > corresponds to the shift-invariant structure. > > The fact that the constant maps to a delta and the successive higher > derivatives to monomials of equally higher order sort of correspond to the > fact that in order to approximate something with such fiendishly local > structure as a delta (corresponding in convolution to taking the value) and > its derivatives (convolution with which is the derivative of corresponding > order) calls for polynomially increasing amounts of high frequency energy. > That is, something you can only handle in the distributional setting, with > its functionals and only a local sense of integration. Trying to interpret > something like that the way we do in conventional L_2 theory sounds likely > to lead to pain. > -- > 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
