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eugene
science forum Guru

Joined: 24 Nov 2005
Posts: 331

Posted: Tue Jun 13, 2006 4:00 pm    Post subject: function with compact support and it's Fourier transform

I ahve the following probably easy question about Foruier transforms:

Let f be a continious function with compact support. Is it true that it's Fourier transform is analytical in R ?

What are the minimal restcrictions on f in order this to be true.

Thanks
victor_meldrew_666@yahoo.
science forum beginner

Joined: 19 May 2006
Posts: 17

Posted: Tue Jun 13, 2006 5:57 pm    Post subject: Re: function with compact support and its Fourier transform

eugene wrote:
 Quote: I ahve the following probably easy question about Foruier transforms: Let f be a continious function with compact support. Is it true that it's Fourier transform is analytical in R ?

Better: its Fourier transform is an analytic function on the whole
complex plane C. (It is easy to justify differentiating under the
this is sufficient to prove holomorphicity).

Victor Meldrew
science forum Guru

Joined: 24 Mar 2005
Posts: 790

Posted: Tue Jun 13, 2006 5:59 pm    Post subject: Re: function with compact support and it's Fourier transform

In article
<28939819.1150214462030.JavaMail.jakarta@nitrogen.mathforum.org>,
eugene <jane1806@mail.ru> wrote:

 Quote: I ahve the following probably easy question about Foruier transforms: Let f be a continious function with compact support. Is it true that it's Fourier transform is analytical in R ? What are the minimal restcrictions on f in order this to be true.

Any L^1 function with compact support has a Fourier transform
that is an entire function. Expand the exponential in the
definition of the FT to see this.
David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Wed Jun 14, 2006 11:01 am    Post subject: Re: function with compact support and it's Fourier transform

On Tue, 13 Jun 2006 12:00:31 EDT, eugene <jane1806@mail.ru> wrote:

 Quote: I ahve the following probably easy question about Foruier transforms: Let f be a continious function with compact support. Is it true that it's Fourier transform is analytical in R ? What are the minimal restcrictions on f in order this to be true.

There are no _minimal_ conditions for this, or if there are nobody
can state them. Compact support is a very common condition, but
for example if |f(x)| <= exp(-|x|) then it also follows that
the fourier transform is analytic on R (in that case it's not
an entire function in the plane, it's holomophic in a certain
horizontal strip.)

 Quote: Thanks

************************

David C. Ullrich

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