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Re: Folding Deltas

• To: mathgroup at smc.vnet.net
• Subject: [mg56940] Re: Folding Deltas
• From: Maxim <ab_def at prontomail.com>
• Date: Tue, 10 May 2005 03:43:16 -0400 (EDT)
• References: <d5muhb$dqs$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On Mon, 9 May 2005 06:04:27 +0000 (UTC), <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)
>

The typesetting is not quite correct: a_i+b_j should be in subscript. You  
can verify the identity (after writing the convolution in the proper way)  
for sums with fixed number of terms using PiecewiseIntegrate from  
http://library.wolfram.com/infocenter/MathSource/5117/ :

In[1]:=
PiecewiseIntegrate[
   (DiracDelta[tau - a1] + DiracDelta[tau - a2])*
     (DiracDelta[x - tau - b1] + DiracDelta[x - tau - b2]),
   {tau, -Infinity, Infinity},
   Assumptions -> a1 < a2]

Out[1]=
DiracDelta[a1 + b1 - x] + DiracDelta[a2 + b1 - x] +
   DiracDelta[a1 + b2 - x] + DiracDelta[a2 + b2 - x]

which just illustrates the fact that convolving with DiracDelta[x - a] is  
equivalent to shifting. Another way:

In[2]:=
% == Integrate[
   (DiracDelta[tau - a1] + DiracDelta[tau - a2])*
     (f[x - tau - b1] + f[x - tau - b2]),
   {tau, -Infinity, Infinity}] /. f -> DiracDelta

Out[2]=
True

Some of the opinions expressed in that essay seem too radical. The point  
about the poor methodology of introducing integrating factors in ODE  
courses is valid, but I'm not so sure about them being useless as such:  
looking for an integrating factor is one of the techniques employed by  
DSolve, which suggests that it covers a reasonably wide number of cases.  
But then I guess one might get into a long argument about how often the  
need for some things implemented in CAS (such as giving symbolic solutions  
of ODEs as huge multi-page expressions involving special functions or  
computing values of those special functions to thousands of digits)  
actually arises in practical applications.

Maxim Rytin
m.r at inbox.ru