= Fourier inversion on a locally compact abelian group
{c}
{title2=$f(x)=\int_{\widehat G}\widehat f(\chi)\chi(x)\,dm_{\widehat G}(\chi)$}
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 $\widehat f\in L^1(\widehat G)$, 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 $\widehat f$ in $L^1$ by <equicontinuity of compact families of characters> and an integrable-tail split.
Back to article page
