Pontryagin dual group ($\hat{G}$)

For a locally compact Hausdorff Abelian group $G$, its Pontryagin dual is the group of continuous unitary characters, with pointwise multiplication and the compact-open topology. Its topology agrees with the Gelfand topology of the L1 convolution algebra. It is again a locally compact Hausdorff Abelian group.

// Target: analysis.bigb

Evaluation $(x,\chi )\mapsto \chi (x)$ is jointly continuous for the compact-open topology on the dual of a locally compact group. One proof uses $\overline{\chi (x)}=\widehat{T_{x}a}(\chi )/\hat{a}(\chi )$ near any character where the denominator is nonzero, together with norm-continuity of translations. A finite neighbourhood cover of a compact family $C$ then gives the displayed uniform convergence. This justifies compact-set-and-integrable-tail estimates even when the group is not metrizable and approximation is indexed by a net.

The dual Haar measure is the scale of Haar measure on the Pontryagin dual group compatible with Fourier inversion on a locally compact abelian group. It can be constructed without first assuming inversion: for a compactly supported test function $\psi$, choose a finite sum of convolution squares $p$ with $\hat{p}>0$ on its support and define $J(\psi )=\int \psi /\hat{p}d{\mu }_p$. The Bochner consistency identity for convolution squares makes this independent of the choice. Modulation by a character shows $J$ is invariant under dual translation; the Riesz representation on compactly supported continuous functions gives the measure. It satisfies $d{\mu }_p=\hat{p}d{m}_{\hat{G}}$ and scales reciprocally when $d{m}_G$ is rescaled.