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Is there something called (Fourier Analysis on infinite groups)? I have read some articles in Fourier analysis on finite groups but I wounder if there is such a theory on infinite groups!

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Yes, see Rudin's book Fourier analysis on groups, for example. The Wikipedia page on Pontryagin duality might be interesting for you. – user50621 Nov 25 '12 at 13:16
@dan : nice link.. thaks :) – Rahman Nov 25 '12 at 13:18
up vote 1 down vote accepted

Fourier analysis of periodic functions is a way to look at functions on $\mathbb{R}/\mathbb{Z}$ (the circle), an infinite group. The transform of such a function is a function on $\mathbb{Z}$, another infinite group. Then you also have the Fourier transform of functions on $\mathbb{R}$, an infinite group. The transform in this case results in a function on $\mathbb{R}$, an infinite group. You'd almost suspect that the discrete Fourier transform is the odd one out. ;-)

And as pointed out by Dan, there is a beautiful theory of Fourier transformations on groups that generalises these examples.

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thanks a lot.. this is very useful :) – Rahman Nov 25 '12 at 13:53

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