I think it can be done simpler. Just extend the inverse Fourier
transform in the same way how the bilateral Laplace transform extends
the direct Fourier transform. Any mistake in that reasoning?
Regards,
Vadim
On 02-Aug-14 20:10, colonel_h...@yahoo.com wrote:
On Fri, 1 Aug 2014, Vadim Zavalishin wrote:
My quick guess is that bandlimited does imply analytic in the complex
analysis sense.
1st off, I am fairly sure it is true that a BL signal cannot be zero
over an interval, so two non-zero BL signals cannot differ by zero over
an interval, so a function with cetain values over any interval is
unique, so the rest of this may be cruft...
However, an audio signal is most often a real valued function of a real
value or a complex valued function of a real value whos imaginary part
happens to be zero (often almost interchangably to little ill effect.)
So to get an analytic complex function you'd have to extend the
function. A non-zero analytic can't have a zero imaginary part, so we'd
need a ``new'' imaginary part and to extent the real and imaginary parts
to a neighborhood of the real line.
Off the cuff I think you might use the real values of f on the real axis
as boundary conditions for the Cauchy-Reimann equations in a
neighborhood of the real axis to solve for a non-zero imaginary part for
f(z) which would then be analytic. This is /if/ BL is enough to show
such a solution exists tehn you're done (which I do not claim is false.
I just can't see a way to get there.)
Ron
--
Vadim Zavalishin
Reaktor Application Architect
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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