Plancherel theorem for locally compact abelian groups

Statement

Let $G$ be an abelian locally compact group with Haar measure $\mu$ , and equip its Pontryagin dual $\widehat G$ with the dual Haar measure $\widehat\mu$ . The Plancherel theorem states that the Fourier transform, initially defined on $L^1(G,\mu)\cap L^2(G,\mu)$ , extends uniquely to a unitary operator between complex Hilbert spaces,

\mathcal F_2:L^2(G,\mu)\longrightarrow L^2(\widehat G,\widehat\mu).

Thus $\|\mathcal F_2f\|_2=\|f\|_2$ , and the inverse unitary is the $L^2$ -extension of inverse Fourier transformation. In particular, the extension preserves Hilbert-space inner products as well as norms.

Meaning of the extension

An arbitrary $L^2$ -function need not be integrable, so its Fourier transform need not be given by a pointwise convergent integral. Choose $f_n\in L^1(G)\cap L^2(G)$ with $f_n\to f$ in $L^2(G)$ ; then $\widehat f_n$ converges in $L^2(\widehat G)$ , and its limit is $\mathcal F_2f$ . Unitarity makes the limit independent of the approximating sequence. The density and extension argument is part of the standard LCA Plancherel theorem Folland, Chapter 4.

Parseval identity and normalization

For $f,g\in L^2(G)$ , unitarity yields

\langle f,g\rangle_{L^2(G)} = \langle \mathcal F_2f,\mathcal F_2g\rangle_{L^2(\widehat G)}.

This Parseval identity includes the norm equality as the case $g=f$ . The theorem also fixes the normalization of $\widehat\mu$ : multiplying $\mu$ by $c>0$ requires multiplying $\widehat\mu$ by $c^{-1}$ .

Standard examples

For $G=\mathbb R^n$ with characters $x\mapsto e^{2\pi i x\cdot\xi}$ , both Haar measures are Lebesgue measure and $\mathcal F_2$ is the usual unitary Euclidean Fourier transform. For $G=\mathbb Z$ with counting measure, the theorem identifies $\ell^2(\mathbb Z)$ unitarily with $L^2(\mathbb T)$ for normalized Haar measure on the circle.

References

Walter Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: Chapter 1, Fourier transformation and the Plancherel theorem.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4, Fourier analysis on locally compact abelian groups.