I want to prove that $T(G)=T(G)*T(G)$ where G is an infinite compact abelian Hausforff Topological group. I'm trying to start this but really im confused with the convolution. Say $f,g \in T(G)$ I need to show that $f*g \in T(G)$, so this is what im getting;

$f*g(x) = \int_G f(sx)g(s^{-1})ds = \int_Gf(s)f(x)g(s^{-1})ds$ but I'm not really sure where to go from there... this whole idea of convolution kind of confuses me and where can you go from $g(s^{-1})$ can we say that this is $\overline{g(s)}$?

I'm also not sure how to approach the other direction... that as trig polynomial can be written as a convolution of two other trig polynomials.


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