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Re: FourierTransform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96190] Re: FourierTransform
  • From: John Doty <jpd at whispertel.LoseTheH.net>
  • Date: Mon, 9 Feb 2009 05:32:20 -0500 (EST)
  • References: <gm1dks$3nk$1@smc.vnet.net> <gm3r8h$mev$1@smc.vnet.net> <gm6kvu$a39$1@smc.vnet.net> <gm99v7$95$1@smc.vnet.net> <200902041021.FAA18709@smc.vnet.net> <gmecf2$adc$1@smc.vnet.net>

Andrzej Kozlowski wrote:
> 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).

Well, the classic of this genre is Berkeley's mocking demolition of 
calculus. Should physicists and engineers have abandoned calculus after 
Berkeley demonstrated that it was pseudo-mathematical nonsense?

One I recall from my undergraduate days was a university computer center 
director who advocated automated scanning of student Fortran jobs on the 
mainframe to detect time-wasting "infinite loops". He was ridiculed as a 
disgrace to his profession: "doesn't he know about the halting 
problem?". But the mockers got it wrong: he wasn't asking for a perfect 
decision procedure, merely a practical one that could weed out the 
easily decidable cases.

Of course, mathematicians perpetrate mathematical howlers, too. Euler's 
assertion that a divergent series may be replaced by its genesis formula 
is an example. The idea that mathematics advances by stepping from truth 
to truth by proving theorems is easily seen to be a myth if you know 
some history. Mathematics is much more interesting, creative, and useful 
than that.

Jens' authority, Vladimirov, asserts that a delta function is only 
defined inside an integral. But I've also heard a mathematician complain 
  "that's not what we mean by integration". And Jens earlier insisted on 
"square integrable" functions. Different notions of "function" and 
"integral" lead to different ideas about what is allowable.

Calculus, Fourier analysis, delta functions, renormalization, all 
recognized as "mathematical nonsense" in their time, yet they proved to 
be indispensable tools for effectively applying mathematics to real 
world problems.

> 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.

The physical design of the communication network you used to broadcast 
this assertion involved heavy use of applied mathematics results 
obtained earlier than this using generalized functions. What is "white 
noise" anyway? Mathematical (at least at the time its use became common) 
and even physical nonsense, but an indispensable concept.

Mathematical objects are products of human imagination. That we can 
obtain reliable knowledge of their properties is a profound mystery. 
That they can effectively model real world objects is another profound 
mystery. But they don't do so perfectly: any particular mathematical 
model of reality will have (often poorly understood) limits to its 
applicability. The justification for any applied mathematics model is 
that you can verify it gets correct answers. The scientific method. The 
problem with most bad applied math in my experience is the dogmatic use 
of inapplicable methods.

> 
> 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 would guess that this sort of formalization is more important to a CAS 
than a human. Physical intuition is reasonably effective at weeding out 
the nonsense here, otherwise these techniques would never have gained a 
foothold. But a CAS has no physical intuition.

It is nice that Mathematica 7 seems to have improved here.

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

Many engineers are suspicious of mathematical abstraction. Mathematical 
notions of "correct" and "true" don't map completely reliably into real 
world situations. So, understanding the connection between the 
abstraction and the physical situation is essential.

> 
> 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
>>
> 
> 


-- 
John Doty, Noqsi Aerospace, Ltd.
http://www.noqsi.com/
--
In theory there is no difference between theory and practice. In 
practice there is. -Yogi Berra


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