Fourier inversion theorem for locally compact abelian groups

Statement

Let $G$ be an abelian locally compact group with Haar measure $\mu$ , and give its Pontryagin dual $\widehat G$ the dual Haar measure $\widehat\mu$ . The Fourier inversion theorem states that if $f\in L^1(G,\mu)$ and its Fourier transform $\widehat f$ belongs to $L^1(\widehat G,\widehat\mu)$ , then

f_0(x)=\int_{\widehat G}\widehat f(\gamma)\gamma(x)\,d\widehat\mu(\gamma)

defines a continuous function vanishing at infinity, and $f_0=f$ almost everywhere. The inverse integral converges absolutely for every $x$ and uses the fixed dual normalization. Hence $f_0(x)=f(x)$ at every point when $f$ itself is continuous.

Why the hypotheses matter

The two $L^1$ assumptions make both transforms ordinary absolutely convergent integrals. Without integrability of $\widehat f$ , inversion may still hold in $L^2$ , by summability, or in the sense of distributions, but the displayed integral need not converge pointwise.

Relation to duality

Applying Fourier transformation on $\widehat G$ and using the evaluation isomorphism $G\cong\widehat{\widehat G}$ from the Pontryagin duality theorem gives

\widehat{\widehat f}(x)=f(-x)

in additive notation under the convention $\widehat f(\gamma)=\int_Gf(x)\overline{\gamma(x)}\,d\mu(x)$ . The reflection changes when the transform convention changes.

Standard examples

For $G=\mathbb R^n$ , the theorem is Euclidean Fourier inversion with the constants determined by the exponential convention. For $G=\mathbb Z$ , it reconstructs an absolutely summable sequence from its Fourier transform on the circle. On a compact abelian group, the dual is discrete and the inverse integral becomes a sum over characters.

References

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