Science > Physics > Does anybody know what's the Fourier transform of the following expression?
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Science > Physics |
| User: |
"Mike" |
| Date: |
18 May 2007 01:23:18 AM |
| Object: |
Does anybody know what's the Fourier transform of the following expression? |
Hi all,
Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir transform
is known to be F(u)?
Is it possible for me to Taylor expand "exp(f(t))" and then sum up the in
frequency domain?
Thanks a lot!
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| User: "Robert Israel" |
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| Title: Re: Does anybody know what's the Fourier transform of the following expression? |
18 May 2007 12:58:57 PM |
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"Mike" <meatheadIV@gmail.com> writes:
Hi all,
Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir transform
is known to be F(u)?
What space are you assuming your functions to be in? Note that if f(t) -> 0
at (+/-)infty, exp(f(t)) -> 1. So at best the Fourier transform of exp(f(t))
will involve a Dirac delta. At worst, exp(f(t)) might not even be a tempered
distribution.
Is it possible for me to Taylor expand "exp(f(t))" and then sum up the in
frequency domain?
I suppose what you mean is exp(f(t)) = sum_{n=0}^infty f(t)^n/n! so
Fourier(exp(f(t))) = sum_{n=0}^infty Fourier(f(t)^n)/n!.
And (formally) Fourier(f(t)^n) is the n'th convolution
power of Fourier(f(t)) (up to a constant, depending on conventions).
I think this should work if f is in the Schwartz space of test
functions, but otherwise there may be problems.
--
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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| User: "Androcles" |
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| Title: Re: Does anybody know what's the Fourier transform of the following expression? |
18 May 2007 02:59:25 AM |
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"Mike" <meatheadIV@gmail.com> wrote in message
news:f2jgok$r0n$1@news.Stanford.EDU...
: Hi all,
:
: Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir
transform
: is known to be F(u)?
Maybe... who is Foureir, anyway?
:
: Is it possible for me to Taylor expand "exp(f(t))" and then sum up the in
: frequency domain?
:
If you could you wouldn't need to ask; so no, it is not possible.
let x = f(t)
http://www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap04/exp-1.jpg
Now it's possible, but whether you can or not I have no way of knowing.
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| User: "Uncle Al" |
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| Title: Re: Does anybody know what's the Fourier transform of the followingexpression? |
18 May 2007 09:48:19 AM |
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Androcles wrote:
"Mike" <meatheadIV@gmail.com> wrote in message
news:f2jgok$r0n$1@news.Stanford.EDU...
: Hi all,
:
: Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir
transform
: is known to be F(u)?
Maybe... who is Foureir, anyway?
[snop]
Fourier, idiot Androclitty. Joseph Fourier of Auxerre. He initiated
the study of hot iron donuts, numbnuts, "Théorie Analytique de la
Chaleur."
Idiot.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
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| User: "Androcles" |
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| Title: Re: Does anybody know what's the Fourier transform of the following expression? |
18 May 2007 11:08:22 AM |
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"Uncle Al" <UncleAl0@hate.spam.net> wrote in message
news:464DBCB3.BA9A56A9@hate.spam.net...
: Androcles wrote:
: >
: > "Mike" <meatheadIV@gmail.com> wrote in message
: > news:f2jgok$r0n$1@news.Stanford.EDU...
: > : Hi all,
: > :
: > : Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir
: > transform
: > : is known to be F(u)?
: >
: > Maybe... who is Foureir, anyway?
: [snop]
I know full well who Fourier was, Uncle Meatball.
You can't spell any better than meathead, can you?
What I asked was who Foureir is.
[snip wet fart]
1) GPS doesn't work for the FAA.
2) Fuckhead.
3) Idiot.
4) Imbecile.
5) Cretin.
6) Arsehole.
7) Biggest tord in the river of *****.
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| User: "Randy Poe" |
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| Title: Re: Does anybody know what's the Fourier transform of the following expression? |
18 May 2007 07:49:38 AM |
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On May 18, 2:23 am, "Mike" <meathea...@gmail.com> wrote:
Hi all,
Is there a Foureir transform for "exp(f(t))" where f(t)'s Foureir transform
is known to be F(u)?
Is it possible for me to Taylor expand "exp(f(t))" and then sum up the in
frequency domain?
You want to write g(t) = sum(i=0,inf) g_i(t) and you are asking
if G(f) = Fourier transform of g(t)
= sum(i=0,inf) G_i(f)
The Fourier transform is linear, so if G(f) and all the
G_i(f) exist, and the Taylor series converges at all t,
then I see no reason you can't do this.
Those are sufficient conditions. Probably a little stronger
than necessary.
- Randy
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