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eugene
science forum Guru
Joined: 24 Nov 2005
Posts: 331
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 Posted: Tue Jun 13, 2006 4:00 pm Post subject:
function with compact support and it's Fourier transform
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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 |
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victor_meldrew_666@yahoo.
science forum beginner
Joined: 19 May 2006
Posts: 17
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| Posted: Tue Jun 13, 2006 5:57 pm Post subject:
Re: function with compact support and its Fourier transform
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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 ?
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Better: its Fourier transform is an analytic function on the whole
complex plane C. (It is easy to justify differentiating under the
integral sign in the Fourier integral under the given hypothesis;
this is sufficient to prove holomorphicity).
Victor Meldrew |
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The World Wide Wade
science forum Guru
Joined: 24 Mar 2005
Posts: 790
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| Posted: Tue Jun 13, 2006 5:59 pm Post subject:
Re: function with compact support and it's Fourier transform
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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. |
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David C. Ullrich
science forum Guru
Joined: 28 Apr 2005
Posts: 2250
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| Posted: Wed Jun 14, 2006 11:01 am Post subject:
Re: function with compact support and it's Fourier transform
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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.)
************************
David C. Ullrich |
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