From: ullrich@math.okstate.edu (David C. Ullrich)
Subject: Re: Band-limited wiggle limit?
Date: Wed, 13 Sep 2000 11:55:40 GMT
Newsgroups: sci.math
Summary: [missing]

On 13 Sep 2000 06:21:18 GMT, baez@galaxy.ucr.edu (John Baez) wrote:

>Ron Hardin wrote:
>
>> What's the most zero crossings you can have in an interval
>> with a band-limited function?
>
>I seem to remember seeing the shocking result that you if you
>fix a certain frequency band [-M,M], M > 0, you can find functions
>whose Fourier transforms lie in that band and take whatever
>values you like at any given finite set of points.  If so,
>there's clearly no limit.
>
>Curiously, I think I saw this result appear as a physics paper
>on the Los Alamos preprint server.  
>
>Does anyone know about this result?  I might be forgetting
>some crucial fine print.

	It's very easy to see that you can get within epsilon of
any values you want on any finite set, so in particular there's
no limit to the number of zero crossings. That's easy enough
that essentially complete proofs have appeared right here
in this thread. (In sci.math.)

	Actually hitting the values you want right on the head
might be harder. Or it could well follow from the within-epsilon
result by successive approximation. (In any case I don't think
it's so shocking - it doesn't seem much more shocking than
the within-epsilon version, and that's almost obvious when
you look at it right.)