From: David C. Ullrich <david_ullrich@my-deja.com>
Subject: Re: Band-limited wiggle limit?
Date: Thu, 07 Sep 2000 15:39:37 GMT
Newsgroups: sci.math
Summary: [missing]

In article <39b7977f.1376283317@nntp.sprynet.com>,
  ullrich@math.okstate.edu wrote:
> On 6 Sep 2000 12:12:49 -0400, kovarik@mcmail.cis.McMaster.CA (Zdislav
> V. Kovarik) wrote:
>
[...]
> >Unless I am missing something in the previous question, the obstacle to
> >having nonzero time-limited functions which wouls also be band-limited is
> >Paley-Wiener Theorem: a band-limited function has an extension to an
> >entire function of exponential type (the easy part of the theorem).
> >
> >Such a function (if nonzero) has only isolated roots which (if there are
> >infinitely many of them) diverge to infinity.
> >
> >(I forgot: is there any estimate of the rate at which they diverge for the
> >entire functions of exponential type?
>
> 	Certainly.
>
> > I would suspect at least linear
> >rate.
>
> 	That seems likely to me as well. It's in those
> books on our shelves (my shelves are a mile away this
> instant.)

      I should learn to wait until I'm awake. Say n(r) is
the number of zeros in {|z| < r} and M(r) is the maximum
of |f| in {|z| <= r}. One tries to apply Jensen's formula
to get a relation between n(r) and M(r) and one doesn't
quite get anything. Duh: Jensen's formula gives a relation
between n(r) and M(2*r). If f(0)=1 then

n(r) <= log(M(2*r)) / log(2) .

In particular if f has exponential type then n(r) grows
at most linearly.

> 	Seems possible that someone could misinterpret
> this. What seems likely to me and I suspect to you as
> well is that there is a linear upper bound on the number
> of zeroes in a disk of radius R. But it's not clear to me
> whether there is actually a specific estimate with a
> specific constant valid for all R > 0 as opposed to just
> saying "O(R) as R->infinity".

      Still not certain whether there's an absolute
bound on the number of zeros in a fixed interval [-a,a]
given that the Fourier transform has support in [-1,1],
say. We get an explicit bound on M(r) in that case, in
terms of (say)

K = the integral from -1 to 1 of |Fourier transform|.

Replacing f by a translate we don't need f(0)=1, we
just need f large at _some_ point of [-a,a], or for that
matter at some point of [-2a, 2a]. But if the sup of |f|
on [-2a,2a] is much smaller than K I don't see where the
explicit estimate comes from.

     (And I begin to think maybe it does not exist:
if f is small everywhere near [-a,a] relative to K
then it should be easier for f to have many zeroes???)

> >)
> >
> >Cheers, ZVK(Slavek)
>
>

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