Fourier inversion on a locally compact abelian group ($f(x)={\int }_{\hat{G}}\hat{f}(\chi )\chi (x)d{m}_{\hat{G}}(\chi )$)

For the dual Haar measure, the formula first follows for continuous compactly supported convolution squares from the Bochner theorem. More generally, if $f\in L^1(G)$ and $\hat{f}\in L^1(\hat{G})$, the right-hand side is a continuous representative of $f$; equality holds everywhere if $f$ is continuous, and otherwise at every point where the chosen representative is continuous. Convolution with normalized small-support positive convolution squares proves the extension: the spatial convolutions approach $f$ in $L^1$, while their transforms approach $\hat{f}$ in $L^1$ by equicontinuity of compact families of characters and an integrable-tail split.