Bochner's theorem for locally compact abelian groups

Statement

Let $G$ be an abelian locally compact group and $\widehat G$ its Pontryagin dual. Bochner's theorem states that a function $\varphi:G\to\mathbb C$ is continuous and positive definite if and only if there is a unique finite positive regular Borel measure $\nu$ on $\widehat G$ such that

\varphi(x)=\int_{\widehat G}\gamma(x)\,d\nu(\gamma) \qquad (x\in G).

Moreover, $\nu(\widehat G)=\varphi(e)$ . The unique $\nu$ is called the representing measure of $\varphi$ ; no Haar normalization is involved. Thus normalized positive-definite functions, characterized by $\varphi(e)=1$ , correspond exactly to probability measures on $\widehat G$ .

Why positivity appears

If $\nu$ is positive, then for $x_1,\ldots,x_n\in G$ and $c_1,\ldots,c_n\in\mathbb C$ ,

\sum_{i,j}c_i\overline{c_j}\varphi(x_j^{-1}x_i) = \int_{\widehat G}\left|\sum_i c_i\gamma(x_i)\right|^2\,d\nu(\gamma)\geq0.

The converse is deeper: positive definiteness produces a cyclic unitary representation, and abelian spectral theory represents its cyclic coefficient by a measure. The equivalence and uniqueness are proved in Folland, Chapter 4.

Standard examples

The constant function $1$ corresponds to the point mass at the trivial character. Every character $\gamma_0$ corresponds to $\delta_{\gamma_0}$ . On $G=\mathbb R^n$ , the theorem is the classical statement that every continuous positive-definite function is the Fourier transform of a finite positive measure.

A continuous function with $\varphi(e)=1$ but a non-positive semidefinite coefficient matrix is a near miss: normalization alone does not produce a positive measure.

Conventions and scope

References

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