Charles Henry escribió:
The error depends on x the signal. Here, I want to make the *convenient* assumption that the spectrum of x is flat, since we want some kind of generality and we want to minimize average error across frequencies. This would make the problem equivalent to using just *one* dirac-delta in place of x and we would get the problem to reduce back to just the difference of the impulse responses
|sinc(t)-f(t)|^2
Ah ok.
This *convenient* assumption is equivalent to (or at least implies) assuming that the only sample that matters for interpolating the signal between -2 and 2 is the one semple at 0. This seems to me a too much strong assumption.
I'm not saying that your conclusion is wrong (though I suspect it is).
Let's take a step back:
Here, I want to make the *convenient* assumption that the spectrum of x is flat
Stated this way, it sounds reasonable, doesn't it. If it does, then it means that by "flat spectrum" you mean the _power spectrum_ of x considered as a _stochastic process_ rather than a deterministic signal.
Brought to the domain of time, assuming x has a flat power spectrum means assuming x is white noise. (btw a closer-to-reality assumption would be that it is pink noise - but that's not the point here) Not a dirac delta.
So minimizing the error would be to minimize the power, or probably energy, of the error meant as a stochastic process.
Though I should have the notions to go a bit further in at least _formulating_ (not solving) the problem, those notions are a bit oxidated, if not completely gone from my head :(
But I'm sure it is not equivalent to minimizing the integral of the difference between the operators applied to a delta function.