Re: Impossible sampling theory!

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 12/10/04 

Date: Fri, 10 Dec 2004 06:23:24 -0600

On Thu, 09 Dec 2004 15:11:54 -0800, Tim Wescott
<tim@wescottnospamdesign.com> wrote:

>David C. Ullrich wrote:
>
>> On Thu, 09 Dec 2004 10:11:07 -0700, Kevin Neilson
>> <kevin_neilson@removethiscomcast.net> wrote:
>> [...]
>>>>
>>>
>>>One of my professors implied that the Dirac delta wasn't mathematically
>>>rigorous but provides correct results so it's used nonetheless.
>>
>>
>> There's no problem with the mathematics involved in the delta
>> function. You don't see the actual math in typical undergraduate
>> courses - the exposition in a typical differential equations
>> book is certainly far from rigorous. That's just because the
>> rigorous explanation is not going to be accessible to that
>> audience, not a problem with the delta function itself.
>> [...]
>
>Can you recommend a good website or book?

If you want the whole story you need to learn some
"real analysis" first (measure theory, topological
vector spaces, etc). There are many places you can
find the theory of distributions worked out in
detail - the two that are standard texts where I
come from would be Folland "Real Analysis" (or
maybe it's "Real Analysis and Applications" or
something) and Rudin "Functional Analysis".

Website? Hmm, google...

It seems that wikipedia

http://en.wikipedia.org/wiki/Distribution

discusses the topic, although I doubt that
there's a complete exposition of the theory
there, in an "encyclopedia" they must have
just statements of the main results.

The description of

http://www.math.ku.dk/~grubb/distcon.pdf

on google sounds like it might be what
you want, but that actual pdf is just a
table of contents. I didn't see how to find
the actual notes on the site, but maybe you
can if you hunt around.

Otoh I wouldn't be surprised if there is no
web site that actually contains the whole story.

************************

David C. Ullrich

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