Bochner's theorem
= Bochner's theorem
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= Bochner theorem
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For a locally compact Hausdorff Abelian group, a continuous <positive-definite function> $p$ is uniquely representable as $p(x)=\int_{\widehat G}\chi(x)\,d\mu_p(\chi)$ for a finite positive <Radon measure> on the <Pontryagin dual group>. Its total mass is $p(e)$. Conversely every such finite positive measure defines a continuous positive-definite function.