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Re: Re: (x-y) DiracDelta[x-y] does not simplify to 0

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• Subject: [mg56345] Re: [mg56297] Re: (x-y) DiracDelta[x-y] does not simplify to 0
• From: Alain Cochard <alain at geophysik.uni-muenchen.de>
• Date: Fri, 22 Apr 2005 06:23:37 -0400 (EDT)
• References: <d42kg5$39t$1@smc.vnet.net> <d45agf$ieu$1@smc.vnet.net> <200504210936.FAA05048@smc.vnet.net> <0e5459acc0a6eae9a16bda863b79434c@mimuw.edu.pl>
• Reply-to: alain at geophysik.uni-muenchen.de
• Sender: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski writes:

 > > I don't understand these reservations. I learned the statement x delta
 > > = 0 in my lectures on distributions at university, and I checked today
 > > that it also appears in one of Laurent Schwartz's ("father" of
 > > distribution theory) books.  Plus I find it very intuitive and it's
 > > straightforward to demonstrate.

 > As I wrote: it takes a lot of interpreting. You not only have to 
 > interpret both x and 0 as distributions, but you also have to interpret 
 > multiplication in a special way, quite different form the normal 
 > Mathematica interpretation. The statement x DiracDelta[x] ==0 is merely 
 > an informal short hand for the staatement:
 > 
 > <x DiracDelta[x], f[x]>= <0, f[x]> for any  "test function" f[x], where 
 > <f,g> stands for Integrate[f[x]*g[x],{-Infinity,Infinity}]
 > 
 > [...]
 > 
 > One can perform certain further informal manipulations on this 
 > 'identity". But such informal manipulations should only be performed 
 > carefully by people who know what they are doing and Mathematica 
 > certainly does not. Because the algebra of distributions is not like 
 > usual "algebra" (for a start multiplication is not defined for 
 > arbitrary distributions) so it is very easy to obtain nonsensical 
 > answers if you manipulate informal expressions containing DiracDelta 
 > blindly, e.g. the following is complete nonsense:
 > 
 > [..] 
 > 
 > [...] since [...] there is actually nothing useful that can be done
 > with the above "relationship", I don't think Mathematica should
 > perform any such "simplifications" except in the proper formal
 > context, which means within Integrate.

At times I feel I understand what you mean, at other times I am not so
sure.

I feel it does not require "a lot" of interpretation.  I'd say it just
requires to be interpreted in the proper sense, but which is the usual
sense of distribution theory (i.e., nothing fancy like I feared before
asking my initial question).

Isn't what you refer to as "shorthand" precisely the way everyone
using distribution theory use distributions all the time?  I'd say
that as far as Mathematica has DiracDelta defined in a way which seems fully
correct to me, it should be one of the "people" who should "know what
they are doing", at least in principle, and detect the nonsensical
cases you are talking about.

So that I better understand, could you please say if you classify the
following relationships in the same class of useless "informal
manipulations" you refer to above.

 UnitStep'[x] = DiracDelta[x] 
 DiracDelta[a x] = DiracDelta[x]/Abs[a] 
 D[DiracDelta[x],x] = DiracDelta'[x]
 x DiracDelta'[x] = -DiracDelta[x]   
 DiracDelta[x-y] = DiracDelta[y-x]

Thanks,
a.

• References:
  • Re: (x-y) DiracDelta[x-y] does not simplify to 0
    • From: Alain Cochard <alain@geophysik.uni-muenchen.de>