This question is probably answered somewhere in SE or math over flow but counldnt find it. Is there a version of Bochner's theorem ( necessary and sufficient condition for positive definiteness) for an arbitrary finite Borel measure, not necessarily positive?
2026-04-09 00:00:41.1775692841
Regarding a theorem of S.Bochner
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An arbitrary (complex) finite Borel measure can be written as $\mu = \mu_1 - \mu_2 + i (\mu_3 - \mu_4)$ where $\mu_j$ are finite positive Borel measures. Corresponding to this, $f$ is the Fourier transform of such a measure iff it can be written as $f = f_1 - f_2 + i (f_3 - f_4)$ where $f_j$ are positive definite functions. In the case of a finite real Borel measure, you don't need the $f_3$ and $f_4$.