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!
Fourier Analysis on infinite groups
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$\begingroup$
group-theory
fourier-analysis
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0Yes, see Rudin's book [Fourier analysis on groups](http://books.google.com/books?id=k3RNtFSBH8gC), for example. The Wikipedia page on [Pontryagin duality](http://en.wikipedia.org/wiki/Pontryagin_duality) might be interesting for you. – 2012-11-25
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0@dan : nice link.. thaks :) – 2012-11-25
1 Answers
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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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0thanks a lot.. this is very useful :) – 2012-11-25