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Re: Re: FourierTransform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96097] Re: [mg96082] Re: FourierTransform
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 5 Feb 2009 04:37:30 -0500 (EST)
  • References: <gm1dks$3nk$1@smc.vnet.net> <gm3r8h$mev$1@smc.vnet.net> <gm6kvu$a39$1@smc.vnet.net> <gm99v7$95$1@smc.vnet.net> <200902041021.FAA18709@smc.vnet.net>

On 4 Feb 2009, at 11:21, John Doty wrote:

> Jens-Peer Kuska wrote:
>> Hi,
>>
>> this is called a distribution or generalized function
>> and not a function and it is only defined
>> inside of an integral as my Vladimirov
>>
>> http://www.amazon.de/Methods-Generalized-Functions-Analytical-Special/dp/0415273560/ref=sr_1_33?ie=UTF8&s=books-intl-de&qid=1233576829&sr=8-33
>>
>> say.
>
> That restriction is Vladimirov's. We who actually apply generalized
> functions to physics and engineering problems are not shy about using
> them outside of integrals. This is the approach that Mathematica
> implements, as you can see below.
>
> A better reference is Bracewell:
> www.amazon.com/Fourier-Transform-Its-Applications/dp/0073039381,
> especially applicable to this question.
>
> By the way, the term "distribution" seems designed to confuse the
> innocent. Many applications of generalized functions also involve
> probability, where the term "distribution" has a different and far  
> more
> familiar meaning.
>
> Most physicists and engineers will drop the "generalized" and simply
> consider things like delta functions to be functions. They often have
> the right properties to represent the behavior of real world objects,
> when other notions of "function" don't.

Sorry, but the last paragraph isn't that much of a recommendation.  
There is hardly any (pseudo)-mathematical nonsense that has not been  
believed to be true or valid by some engineer,  occasionally with  
lamentable consequences. I remember that when, I was a math  
undergraduate, a "popular" mathematics journal published a long list  
of examples (with detailed references) of mathematical nonsense  
perpetrated by engineers, economists and some others (some quite  
hilarious). I think some of it might have involved the use (or misuse)  
of generalized functions, though I am no longer sure. (I guess I could  
still trace the list, if you really wanted to see it ...but you could  
instead just search the archives of this forum ;-))

The most intuitive yet rigorous theory of generalized functions that  
allows then to be treated as "functions" was created by Jean Francois  
Colombeau. He wrote a beautifully clear and quite short exposition of  
his theory entitled "Elementary Introduction to new Generalized  
Functions" (North-Holland 1985).
The Colombeau genealized functions have values at all points, but they  
are "generalized numbers" (which include ordinary numbers). Thus the  
Dirac Delta is a function defined on R, whose value is 0 for x!=0 and  
a generalized number at x==0. Colombeau generalized functions have  
derivatives of all orders (which are themselves generalized functions)  
and classical distributions are precisely those generalized functions  
which, in a neighborhood of each point, are partial derivatives of  
continuous functions.
Colombeau theory was the first one that solved satisfactorily the old  
problem of giving a satisfactory definition of multiplication of  
generalized functions (something that before Colombeau was considered  
impossible by many). Colembeau theory justified much of the heuristics  
that had been previously known to physicists but it also helped to  
uncover nonsense where there was nonsense to uncover.

Last but not least, Mathematica's notion of a generalized function is  
based on Colombeau. To convince yourself look at the documentation for  
HeavisideTheta, in the section "Possible issues". I quote:

Products of distributions with coincident singular support cannot be  
defined (no Colombeau algebra interpretation).

I get the impression that not all engineers believe in this even now.

Andrzej Kozlowski


>
>
>>
>> Regards
>>   Jens
>>
>> John Doty wrote:
>>> Jens-Peer Kuska wrote:
>>>> Hi,
>>>>
>>>> the Fourier transform over the interval x in (-Infinity,Infinity)
>>>> converges only for quadratic integrable functions, i.e., functions
>>>> where Integrate[Conjugate[f[x]]*f[x],{x,-Infinity,Infinity}]<  
>>>> Infinity
>>>>
>>>> This is not the case for Cosh[x], and so no Fourier transform  
>>>> exist.
>>> Depends on what you mean by "function". Mathematica tries in its
>>> pragmatic way to do what you might want here:
>>>
>>> In[1]:= FourierTransform[t^2,t,w]
>>>
>>> Out[1]= -(Sqrt[2 Pi] DiracDelta''[w])
>>>
>>> t^2 is certainly not square integrable, but this is the kind of  
>>> useful
>>> result scientists and engineers want.
>>>
>>> Mathematica's support for "generalized functions" still has room for
>>> improvement, but it has come a long way. The bizarre problems I  
>>> saw in
>>> the past trying Fourier methods to perform fractional  
>>> differentiation
>>> and integration
>>> (http://forums.wolfram.com/mathgroup/archive/2000/Apr/msg00043.html)
>>> seem no longer to be with us in Mathematica 7.
>>>
>>
>
>
> -- 
> John Doty, Noqsi Aerospace, Ltd.
> http://www.noqsi.com/
> --
> The axiomatic method of mathematics is one of the great achievements  
> of
> our culture. However, it is only a method. Whereas the facts of
> mathematics once discovered will never change, the method by which  
> these
> facts are verified has changed many times in the past, and it would be
> foolhardy to expect that changes will not occur again at some future
> date. - Gian-Carlo Rota
>



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