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Re: FourierTransform

  • To: mathgroup at
  • Subject: [mg96037] Re: FourierTransform
  • From: John Doty <jpd at>
  • Date: Mon, 2 Feb 2009 06:21:40 -0500 (EST)
  • References: <gm1dks$3nk$> <gm3r8h$mev$>

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 
seem no longer to be with us in Mathematica 7.

John Doty, Noqsi Aerospace, Ltd.
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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