On 2015-06-29, vadim.zavalishin wrote:
Roughly (or maybe even strictly?) speaking, the sums converge if and
only if the derivatives fall off sufficiently fast, which seems to be
equivalent to the relaxed (exponential growth on the complex plane,
instead of exponential growth on the imaginary axis and polynomial
growth on the real axis) Paley-Wiener-Schwartz conditions.
I'm still going to have to let this one lie for awhile, because I don't
think it's what Schwarz's theorem actually says. Or that of Hörmander.
But let's see... :)
I strongly believe it does make sense. Because you want to distinguish
between the waveforms which can be antialiased by BLEP (at least
theoretically) and those which can not.
But if you take a look at where I was going towards the end of my post,
it might be that a sufficiently complicated BLEP derivation actually
leads to a form of interpolation which pretty much handles it all, to
any finite order of polynomial interpolation and discontinuities. If
that holds, the class of things it can handle is actually dense within
the class of tempered distributions. Thus, basically "anything".
For those which can the practical error of antialiasing will be
determined by the truncation of the BLEP sum and the truncation of the
BLEPs themselves (windowing in time domain).
Okay, for practical applications, so the reasoning goes, because you
won't be stacking on quite the amount of baggage I circumscribed.
But in any case tell me, does what I was talking about seem like your
intuition? What do you think is the part which needs the most work
still?
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