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Re: FourierTransform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96082] Re: FourierTransform
  • From: John Doty <jpd at whispertel.LoseTheH.net>
  • Date: Wed, 4 Feb 2009 05:21:40 -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>

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.

> 
> 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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