Bochner's theorem

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.

Bochner's theorem for a locally compact abelian group G, with dual group ${\displaystyle {\widehat {G}}}$, says the following:

Theorem For any normalized continuous positive definite function f on G (normalization here means f is 1 at the unit of G), there exists a unique probability measure on ${\displaystyle {\widehat {G}}}$ such that

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