Inversion of Fourier Transformation

As we know, the inversion formula of Fourier transformation holds pointwise for Schwartz class. We also have a general result concerning the inversion of Fourier transformation on locally compact abelian groups, which says that if $f$ belongs to the intersection of the $L^1$-algebra and the Fourier-Stieltjes algebra on a locally compact abelian group $G$, then the inversion formula holds a.e. for $f$. And the above result can be generalized in special cases. For example, If $G$ is $R$ or $R/Z$, the Carleson-Hunt theorem says the inversion formula holds a.e. for $f$ in $L^p$ with $1<p<\infty$.

My question is, is there any other version of generalization of inversion of Fourier transformation concerning a given locally compact abelian group $G$? For example, $G$ is an abelian Lie group, or $G$ is a compact group?

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I am out of the office so can't look up precise references right now, but try googling or MathSciNet-searching for "Feichtinger's Segal algebra" or some similar phrase. This is a construction which attempts to find for a general LCA group an algebra that is well-behaved under Fourier transform in a similar fashion to the Schwartz class on $R^n$, and hence might be along the lines you are looking for. – Yemon Choi Mar 24 2011 at 7:38
P.S. in your last sentence you presumably want to restrict attention to compact abelian groups -- Fourier analysis on nonabelian compact Lie groups has been much studied but opens up a whole new can of worms – Yemon Choi Mar 24 2011 at 7:39
Thanks, I'll sesrch for the relevant information. – Acky Mar 24 2011 at 8:17