On 2015-06-19, Ethan Duni wrote:
Not exactly. If you take the typical sampling formula, with
equidistant samples, you need them all.
Yeah, that's what we're discussing isn't it?
Are we? You can approximate any L_2 bandlimited function arbitrarily
closely with a finite number of samples. I don't think you can even
approach a polynomial in the distributional sense absent the whole
infinite set.
But in theory pretty much any numerable number of samples from any
compact interval will do.
Sure, but that's not going to help us with figuring out what comes out
of an audio DAC.
Yes. And I'm sorry if I sound of like a know-it-all or show-off here. I
really am anything *but*. Just interested in this stuff. :)
Linearly? It dies off as 1/x.
Yeah that's what I mean. Kind of informal, but "die off" was meant to
imply "this is what is in the denominator."
Check. But 1/x is still pretty special in the denominator.
Not quite so. The proper way to say it is "when probed locally by
nice enough test functions, the reconstruction works the same".
I'm not sure we're on the same page here - the statement you were
replying to was referring to the classical L2 sampling theorem stuff.
If so, again sorry. I have tried to work as much in the distributional
setting as I can.
The sinc convolution is just fine even in this setting.
??? The sinc convolution is not implementable in any setting.
It actually is in the distributional setting. When you go via the weak*
topology of the relevant functional space, the functions they induce
back implement pure sinc interpolation. The limit is exact.
But yeah, in *reality* nothing of the sort can exist. You just have to
approximate. It's just that there's nothing new there for any of us, I
think. Delta-sigma, yadda-yadda, it's what them chips do all the time
for us. Right? ;)
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