Re: Fourier transform of a compact C_inf function

On Mon, 31 Aug 2009 16:08:34 -0500, David C Ullrich wrote:

On Mon, 31 Aug 2009 22:19:28 +0200, bubu wrote:

Hi,

Paley-wiener explain which function is Fourier Transform of a compact
support, C_inf function.


a function f(z) is Fourier Transform of a compact support C_inf function
iff:
- f(z) is entire
- f(z) is L2 on horizontal lines => int(abs(f(x+i*y))^2,x=-inf..inf)<inf
- f(z) is of exponential type => abs(f(z))<=k*exp(A*abs(z))

There are various "Paley-Wiener" theorems. The conditions above
are for the Fourier transform of an L^2 function with compact
support, not a C^infinity function. To get C^infinity you add
the condition

(1+|x|)^d |f(x)| <= c_d

for every d.

I'm looking this sort of function. i tried sin(z)/z, J(n,z), Kummer
function but i always try ...

sin(z)/z satisfies all the conditions you listed, but
not the one I added. I don't know if you should expect to
find a simple explicit formula for a function with all
four conditions...

Not in "closed form", no, but you could try something like

f(z) = int_{-1}^1 exp(1/(x^2-1) + i x z) dx

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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