Question

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?

Answer 1

The analogues of Schwartz functions on general locally compact abelian groups are called Schwartz-Bruhat functions, and are mapped to Schwartz-Bruhat functions under Fourier transforms. Their dual spaces are spaces of tempered distributions on such groups which are mapped to other tempered distributions under Fourier transforms. The tempered distributions on these groups include most functions you might want to take a Fourier transform of.