Bochner's theorem on Fourier transforms appeared in a 1932 book.

In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series.

Bochner's theorem.

Bochner's theorem characterizes which functions may arise as the Fourier-Stieltjes transform of a measure .

The positive definiteness of this single-argument version of the covariance function can be checked by Bochner's theorem.

One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context.

Bochner's theorem for a locally compact Abelian group G, with dual group , says the following:

The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.

The central result here is Bochner's theorem, although its usefulness is limited because the main condition of the theorem, non-negative definiteness, is very hard to verify.

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.

If is continuous, Bochner's theorem can be used to prove that its Fourier transform exists as a positive measure, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).