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Theorem

Bochner's Theorem

Analysis

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
Statement
Bochner'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
Proof Year
1932 2
Classification
Statement Form
Characterization Theorem 1
Connections

Has Statement Form

Characterization, Concepts

In Branch

Harmonic Analysis, Branches of Mathematics
Source Bochner's Theorem (Wikipedia)

Proved By

Gustav Herglotz, Mathematicians
Source Bochner's Theorem (Wikipedia)
Salomon Bochner, Mathematicians
Source Bochner's Theorem (Wikipedia)
Sources
1. Bochner's Theorem (Wikipedia)
Wikimedia Foundation
  • 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.
  • In Branch: Harmonic Analysis, Lead paragraph
    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.
  • Proved By: Salomon Bochner, Lead
    In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real line
  • Proved By: Gustav Herglotz, Lead
    The case of sequences was first established by Gustav Herglotz
View the Source
2. Salomon Bochner (Wikipedia)
Wikimedia Foundationbiography, paragraph on his Fourier transform work
Quote, biography, paragraph on his Fourier transform work
Bochner's theorem on Fourier transforms appeared in a 1932 book.
View the Source
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