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Re: Re: Re: Folding Deltas
It's strange how some pages disappear quickly (
http://xoom.virgilio.it/Home.php ), but at least archive.org can come
to the rescue:
http://web.archive.org/web/20040615001739/http://xoomer.virgilio.it/maurocer/Text07.htm
On 5/11/05, Daniel Lichtblau <danl at wolfram.com> wrote:
> Zhengji Li wrote:
> > Integrate[DiracDelta[t], {t, -a, a}] = 1, where a > 0.
> >
> > Maybe Mathematica think Integrate[DiracDelta[t] DiracDelta[t - 2] ,
> > {t, -3, 3}] is a little bit complicated. (As far as I know, the result
> > should be 0)
> >
> > But, Integrate[Anything, {t, a, b}] + Integrate[Anything, {t, b, a}] =
> > 0, so you will get the result.
> >
> > On 5/9/05, baermic at yahoo.com <baermic at yahoo.com> wrote:
> >
> >>Can anyone help to verify in Mathematica the expression given by Rota
> >>(http://xoomer.virgilio.it/maurocer/Text07.htm):
> >>
> >>Convolution ( Sum of DiracDeltaFct ** Sum of DiracDeltaFct) == Sum
> >>(DiracDeltaFct + Values).
> >>
> >>I tied
> >>
> >>Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, -3, 3} ]
> >>which does not evaluate;
> >>but
> >>
> >>Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, -3, 1} ] +
> >>Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, 1, 3} ] == 0
> >>True
> >>
> >>( I use ver 5.1 with W2k)
> >>
>
>
> Actually the product of delta functions is not defined. This is because
> it is not possible to make it well defined. For example I could give
> limiting approximations to the integrand that make that integral give
> any result, not just zero. There are also other ways to show it is not
> well defined.
>
> The original note was a bit unclear but I the correct form of the
> identity in question uses convolution of deltas, not product. Also there
> is a minor typo at the web page for Rota's talk. The idenity boils down
> simply to
>
> delta(a_i) * delta(b_j) = delta(a_i+b_j)
>
> where '*' denotes convolution, not ordinary product. This follows from
> basic rules of convolution, translation, and the fact that delta(0) is
> the identity element for convolution.
>
>
> Daniel Lichtblau
> Wolfram Research
>
>
>
>
--
Chris Chiasson
http://chrischiasson.com/
1 (810) 265-3161
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