Bochner's theorem

This article is about Bochner's theorem in harmonic analysis. For Bochner's theorem in Riemannian geometry, see Bochner's theorem (Riemannian geometry).

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. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.)[1]

  1. ^ William Feller, Introduction to probability theory and its applications, volume 2, Wiley, p. 634