Dual Haar measure - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group that is abelian, with Haar measure $\mu$ , and let $\widehat G$ be its Pontryagin dual. The dual Haar measure $\widehat\mu$ is the unique Haar measure on $\widehat G$ for which the Fourier transform

\widehat f(\gamma)=\int_G f(x)\overline{\gamma(x)}\,d\mu(x)

extends from $L^1(G,\mu)\cap L^2(G,\mu)$ to a unitary operator

L^2(G,\mu)\longrightarrow L^2(\widehat G,\widehat\mu).

Thus $\widehat\mu$ is not an independent normalization: it is determined by $\mu$ and the chosen Fourier-transform convention.

Fourier inversion

The Fourier inversion theorem says that, with the dual normalization, sufficiently integrable functions satisfy

f(x)=\int_{\widehat G}\widehat f(\gamma)\gamma(x)\,d\widehat\mu(\gamma).

More precisely, if $f\in L^1(G)$ and $\widehat f\in L^1(\widehat G)$ , the right side gives a continuous representative equal to $f$ almost everywhere. This inversion criterion and the unitary $L^2$ extension in the Plancherel theorem determine the same normalization Rudin, Chapter 1.

Scaling and bidual normalization

If $\mu$ is replaced by $c\mu$ for $c>0$ , then $\widehat\mu$ must be replaced by $c^{-1}\widehat\mu$ . Applying the construction again on $\widehat G$ , and identifying $G$ with its bidual through Pontryagin duality, recovers the original measure $\mu$ .

Standard normalizations

For $G=\mathbb R^n$ with Lebesgue measure and characters $x\mapsto e^{2\pi i x\cdot\xi}$ , the dual measure is Lebesgue measure. For $G=\mathbb Z$ with counting measure, the dual circle has normalized Haar probability measure. Conversely, normalized Haar measure on the circle is dual to counting measure on $\mathbb Z$ .

References

W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. DOI record. Relevant: Haar measure on the dual group, inversion, and Plancherel theory.
G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4, dual measures and Fourier analysis on locally compact abelian groups.