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.

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## The theorem for locally compact abelian groups

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

**Theorem** For any normalized continuous positive definite function *f* on *G* (normalization here means *f* is 1 at the unit of *G*), there exists a unique probability measure on such that

i.e. *f* is the Fourier transform of a unique probability measure μ on . Conversely, the Fourier transform of a probability measure on is necessarily a normalized continuous positive definite function *f* on *G*. This is in fact a one-to-one correspondence.

The Gelfand-Fourier transform is an isomorphism between the group C*-algebra C*(*G*) and C_{0}(*G*^). 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 *G* (the proof in fact shows every normalized continuous positive definite function must be of this form).

Given a normalized continuous positive definite function *f* on *G*, one can construct a strongly continuous unitary representation of *G* in a natural way: Let *F*_{0}(*G*) be the family of complex valued functions on *G* with finite support, i.e. *h*(*g*) = 0 for all but finitely many *g*. The positive definite kernel *K*(*g*_{1}, *g*_{2}) = *f*(*g*_{1} - *g*_{2}) induces a (possibly degenerate) inner product on *F*_{0}(*G*). Quotiening out degeneracy and taking the completion gives a Hilbert space

whose typical element is an equivalence class [*h*]. For a fixed *g* in *G*, the "shift operator" *U _{g}* defined by (

*U*)(

_{g}*h*) (g') =

*h*(

*g' - g*), for a representative of [

*h*], is unitary. So the map

is a unitary representations of *G* on . By continuity of *f*, it is weakly continuous, therefore strongly continuous. By construction, we have

where [*e*] is the class of the function that is 1 on the identity of *G* and zero elsewhere. But by Gelfand-Fourier isomorphism, the vector state on C*(*G*) is the pull-back of a state on , which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

On the other hand, given a probability measure μ on , the function

is a normalized continuous positive definite function. Continuity of *f* follows from the dominated convergence theorem. For positive definiteness, take a nondegenerate representation of . This extends uniquely to a representation of its multiplier algebra and therefore a strongly continuous unitary representation *U _{g}*. As above we have

*f*given by some vector state on

*U*

_{g}therefore positive-definite.

The two constructions are mutual inverses.

## Special cases

Bochner's theorem in the special case of the discrete group **Z** is often referred to as Herglotz's theorem, (see Herglotz representation theorem) and says that a function *f* on **Z** with *f*(0) = 1 is positive definite if and only if there exists a probability measure μ on the circle **T** such that

Similarly, a continuous function *f* on **R** with *f*(0) = 1 is positive definite if and only if there exists a probability measure μ on **R** such that

## Applications

In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables of mean 0 is a (wide-sense) stationary time series if the covariance

only depends on *n*-*m*. The function

is called the autocovariance function of the time series. By the mean zero assumption,

where ⟨⋅ , ⋅⟩ denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that
*g* is a positive definite function on the integers ℤ. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that

- .

This measure μ is called the **spectral measure** of the time series. It yields information about the "seasonal trends" of the series.

For example, let *z* be an *m*-th root of unity (with the current identification, this is 1/m ∈ [0,1]) and *f* be a random variable of mean 0 and variance 1. Consider the time series . The autocovariance function is

- .

Evidently the corresponding spectral measure is the Dirac point mass centered at *z*. This is related to the fact that the time series repeats itself every *m* periods.

When *g* has sufficiently fast decay, the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative *f* is called the spectral density of the time series. When *g* lies in *l*^{1}(ℤ), *f* is the Fourier transform of *g*.

## See also

## References

- Loomis, L. H. (1953),
*An introduction to abstract harmonic analysis*, Van Nostrand - M. Reed and B. Simon,
*Methods of Modern Mathematical Physics*, vol. II, Academic Press, 1975. - Rudin, W. (1990),
*Fourier analysis on groups*, Wiley-Interscience, ISBN 0-471-52364-X