Re: FourierTransform
- To: mathgroup at smc.vnet.net
- Subject: [mg96037] Re: FourierTransform
- From: John Doty <jpd at whispertel.LoseTheH.net>
- Date: Mon, 2 Feb 2009 06:21:40 -0500 (EST)
- References: <gm1dks$3nk$1@smc.vnet.net> <gm3r8h$mev$1@smc.vnet.net>
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