Band limited function zero crossings, how many?

Matt Timmermans wrote:
> "Robert E. Beaudoin" <reb@sens.com> wrote in message
> news:42c6a$40369899$44a72252$11987@msgid.meganewsservers.com...
> 
[snip]
>>I wonder just which functions on [-1, 1] have extensions in L^2(R)
>>whose Fourier transforms are zero outside [-W, W] (for a given W).
>>I also wonder if there is a bound (for W fixed, of course) on the
>>number of zero-crossings in [-1, 1] such a function can have; that
>>would address the OP's OQ (original question).  My guess is that
>>there is, but I need to think about this some more....
> 
> 
> We can certainly rule out a bound on the number of zero crossings, given
> that the Nth PSWF *has* N zeros in the interval.  Given how fast the
> concentrations fall off for successive PSWFs, though, I think you will fail
> to get convergence on L^2(R) for a lot of interesting functions -- even
> smooth ones.
> 
> 

The sound you hear is me smacking my head; how'd I miss that?  Thanks
for pointing it out and settling the OP's OQ.  Definitely more than
smoothness is required of the restriction of a bandlimited function
to [-1, 1] (say), as every bandlimited function has an extension to
an entire function.  The Paley-Wiener theorem even characterizes
which entire functions can arise this way (at least for bandlimited
functions with smooth spectra; I think there are extensions of the
theorem that apply when the spectrum is not smooth, too) which sorta
tells you which functions on [-1, 1] have bandlimited extensions
(the ones with entire extensions satisfying the Paley-Wiener bounds,
of course) but this is a pretty opaque characterization.  Maybe
there is some way to simplify it (or maybe not).

Well, I hope that was of interest.  Gotta go smack my head some more....

Bob Beaudoin

Fred Marshall wrote:
...
> (I'm not sure everyone knows what "complete" means.....)

From the paper Matt posted, I gather it means what we in German refer
to as "dicht" (dense):

Set A is "dicht" in set B, if for all b \in B there exists a sequence
a_i \in A, such that a_i -> b, i-> \infty. (Note: you need a metric
space M, with A \subset B \subset M to define what convergence means).

For example, the rational numbers are "dicht" in the real numbers.

Is that what "complete" means?

 
> 
> Fred

I wrote:

> Fred Marshall wrote:
> ...
> > (I'm not sure everyone knows what "complete" means.....)
> 
> From the paper Matt posted, I gather it means what we in German refer
> to as "dicht" (dense):
> 
> Set A is "dicht" in set B, if for all b \in B there exists a sequence
> a_i \in A, such that a_i -> b, i-> \infty. (Note: you need a metric
> space M, with A \subset B \subset M to define what convergence means).
> 
> For example, the rational numbers are "dicht" in the real numbers.
> 
> Is that what "complete" means?

No one answered, so I looked it up. It's close to what I thought, but
the term "complete" is appearantly reserved for function spaces, and
refers specifically to convergence to piecewise continuous functions
in the L2-norm:

http://mathworld.wolfram.com/CompleteOrthogonalSystem.html

Piecewise continuous functions are "dicht" in the set of measurable
functions (and thus L2-functions), so this is a very broad class of
functions.

Regards,
Andor