Re: Re: Folding Deltas

*To*: mathgroup at smc.vnet.net*Subject*: [mg56967] Re: [mg56906] Re: [mg56876] Folding Deltas*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Wed, 11 May 2005 05:24:43 -0400 (EDT)*References*: <200505090545.BAA13779@smc.vnet.net> <200505100742.DAA08259@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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

**Follow-Ups**:**Re: Re: Re: Folding Deltas***From:*Chris Chiasson <chris.chiasson@gmail.com>

**References**:**Folding Deltas***From:*baermic@yahoo.com

**Re: Folding Deltas***From:*Zhengji Li <zhengji.li@gmail.com>