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.
)[1] Bochner's theorem for a locally compact abelian group
, says the following: Theorem For any normalized continuous positive-definite function
), there exists a unique probability measure
is the Fourier transform of a unique probability measure
Conversely, the Fourier transform of a probability measure on
is necessarily a normalized continuous positive-definite function
The theorem is essentially the dual statement for states of the two abelian C*-algebras.
The proof of the theorem passes through vector states on strongly continuous unitary representations of
(the proof in fact shows that every normalized continuous positive-definite function must be of this form).
, one can construct a strongly continuous unitary representation of
Quotienting out degeneracy and taking the completion gives a Hilbert space
whose typical element is an equivalence class
, which is necessarily integration against a probability measure
This extends uniquely to a representation of its multiplier algebra
Bochner's theorem in the special case of the discrete group
In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series.
of mean 0 is a (wide-sense) stationary time series if the covariance
is called the autocovariance function of the time series.
denotes the inner product on the Hilbert space of random variables with finite second moments.
By Bochner's theorem, there exists a unique positive measure
is called the spectral measure of the time series.
It yields information about the "seasonal trends" of the series.
-th root of unity (with the current identification, this is
Evidently, the corresponding spectral measure is the Dirac point mass centered at
This is related to the fact that the time series repeats itself every
has sufficiently fast decay, the measure
is absolutely continuous with respect to the Lebesgue measure, and its Radon–Nikodym derivative
is called the spectral density of the time series.