Fourier transform on a locally compact abelian group

Definition

Let $G$ be a locally compact group that is abelian, let $\widehat G$ be its Pontryagin dual, and fix a Haar measure $\mu$ on $G$ . For an $L^1$ function $f$ on $G$ , its Fourier transform is the function on $\widehat G$ defined by

\widehat f(\gamma) = \int_G f(x)\overline{\gamma(x)}\,d\mu(x), \qquad \gamma\in\widehat G.

The conjugate specifies the sign convention; using $\gamma(x)$ gives the inverse convention. The transform is bounded and continuous and vanishes at infinity.

Convolution and translation

For $f,g\in L^1(G)$ , the convolution theorem states

\widehat{f*g}(\gamma)=\widehat f(\gamma)\widehat g(\gamma).

Thus the Fourier transform converts the group-defined convolution product into pointwise multiplication on the dual group. Translation by $y\in G$ becomes multiplication by the character value $\overline{\gamma(y)}$ .

Inversion and Plancherel normalization

There is a unique normalization of Haar measure $\widehat\mu$ on $\widehat G$ with the following property: if $f\in L^1(G)$ and $\widehat f\in L^1(\widehat G)$ , then $f$ agrees almost everywhere with the continuous function

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

With the same normalization, the transform extends uniquely from $L^1(G)\cap L^2(G)$ to a unitary map

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

which is the Plancherel theorem for locally compact abelian groups.

Rescaling $\mu$ rescales $\widehat f$ , so the dual Haar measure must be normalized correspondingly.

Standard models

For $G=\mathbb R^n$ and characters $\gamma_\xi(x)=e^{2\pi i x\cdot\xi}$ , the definition is the classical Euclidean Fourier transform with exponent $-2\pi i x\cdot\xi$ . For $G=\mathbb Z$ , the dual is the circle and the transform produces a Fourier series. If $G$ is compact, $\widehat G$ is discrete; if $G$ is discrete, $\widehat G$ is compact.

The general LCA construction, inversion, and duality are treated in Rudin, the opening chapters.

References

Walter Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: the opening chapters on LCA groups, Fourier transforms, inversion, and duality.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. Routledge publisher record. Relevant: Chapter 4, “Analysis on Locally Compact Abelian Groups.”