Bochner's Theorem characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real line, identifying exactly which functions can arise this way. More generally in harmonic analysis, it states that a continuous positive-definite function on a locally compact abelian group corresponds, under the Fourier transform, to a finite positive measure on the group's Pontryagin dual. The theorem's special case for sequences was first established by Gustav Herglotz.
Facts
StatementBochner's theorem characterizes which continuous positive definite functions on a locally compact abelian group are Fourier transforms of a finite positive measure on the group's dual, giving a one to one correspondence between such functions and such measures. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Bochner's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentenceQuote, lead paragraph, first sentence
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real line.
2. Salomon Bochner (Wikipedia)
Wikimedia Foundationbiography, paragraph on his Fourier transform workQuote, biography, paragraph on his Fourier transform work
Bochner's theorem on Fourier transforms appeared in a 1932 book.
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