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. The general theorem and its normalization are developed in Rudin, Chapter 1.

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.