On 2015-06-19, Ethan Duni wrote:
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.
In fact even that survives fully. In the local integration framework that the tempered distributions carry with them, you can convolve a polynomial by a delta function or any finite derivative of it, and you can also apply a Dirac comb to it so as to sample it.
But what does the convergence of the Shannon-Whittaker formula look like in the case of stuff like polynomials?
Precisely the same as it does in the case of any other function. You just have to take the convergence in the weak* sense, and then do some extra legwork to return that into a function, from the functional domain. What it returns to is precisely the unique polynomial (or whatnot) you're after. The reconstruction formula, using sinc functions, is exact in that circuitous sense.
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.
All that is taken care of by the fact that the reconstruction is defined as a transposition of a functional wrt the Schwartz space to begin with. All the mechanics are local because of that. The asymptotics don't matter after that, and the Shannon-Whittaker formula is suddenly defined locally, so that growth rates upto polynomial don't matter at all.
Of course some funky global, dual shit happens then: you actually need all of the samples from -inf to +inf in order to define any polynomial, and no finitely supported in time subset will suffice.
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