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