Bochner's theorem explained

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. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.)

The theorem for locally compact abelian groups

, with dual group , says the following:

Theorem For any normalized continuous positive-definite function

on (normalization here means that is 1 at the unit of ), there exists a unique probability measure on such that

$$f(g)\; =\; \backslash int\_\; \backslash xi(g)\; \backslash ,d\backslash mu(\backslash xi),$$

i.e.

is the Fourier transform of a unique probability measure on . Conversely, the Fourier transform of a probability measure on is necessarily a normalized continuous positive-definite function on . This is in fact a one-to-one correspondence. and . 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).

Given a normalized continuous positive-definite function

on , one can construct a strongly continuous unitary representation of in a natural way: Let be the family of complex-valued functions on with finite support, i.e. for all but finitely many . The positive-definite kernel induces a (possibly degenerate) inner product on . Quotienting out degeneracy and taking the completion gives a Hilbert space

$$(\backslash mathcal,\; \backslash langle\; \backslash cdot,\; \backslash cdot\backslash rangle\_f),$$

whose typical element is an equivalence class

. For a fixed in , the "shift operator" defined by , for a representative of , is unitary. So the map

$$g\; \backslash mapsto\; U\_g$$

is a unitary representations of

on . By continuity of , it is weakly continuous, therefore strongly continuous. By construction, we have

$$\backslash langle\; U\_g\; [e],\; [e]\; \backslash rangle\_f\; =\; f(g),$$

where

is the class of the function that is 1 on the identity of and zero elsewhere. But by Gelfand–Fourier isomorphism, the vector state on is the pull-back of a state on , which is necessarily integration against a probability measure . Chasing through the isomorphisms then gives

$$\backslash langle\; U\_g\; [e],\; [e]\; \backslash rangle\_f\; =\; \backslash int\_\; \backslash xi(g)\; \backslash ,d\backslash mu(\backslash xi).$$

On the other hand, given a probability measure

on , the function

$$f(g)\; =\; \backslash int\_\; \backslash xi(g)\; \backslash ,d\backslash mu(\backslash xi)$$

is a normalized continuous positive-definite function. Continuity of

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 . As above we have given by some vector state on

$$f(g)\; =\; \backslash langle\; U\_g\; v,\; v\; \backslash rangle,$$

therefore positive-definite.

The two constructions are mutual inverses.

Special cases

is often referred to as Herglotz's theorem (see Herglotz representation theorem) and says that a function on with is positive-definite if and only if there exists a probability measure on the circle such that

$$f(k)\; =\; \backslash int\_\; e^\; \backslash ,d\backslash mu(x).$$

Similarly, a continuous function

on with is positive-definite if and only if there exists a probability measure on such that

$$f(t)\; =\; \backslash int\_\; e^\; \backslash ,d\backslash mu(\backslash xi).$$

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

$$\backslash operatorname(f\_n,\; f\_m)$$

only depends on

. The function

$$g(n\; -\; m)\; =\; \backslash operatorname(f\_n,\; f\_m)$$

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

$$g(n\; -\; m)\; =\; \backslash langle\; f\_n,\; f\_m\; \backslash rangle,$$

where

denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that is a positive-definite function on the integers . By Bochner's theorem, there exists a unique positive measure on such that

$$g(k)\; =\; \backslash int\; e^\; \backslash ,d\backslash mu(x).$$

This measure

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

For example, let

be an -th root of unity (with the current identification, this is ) and be a random variable of mean 0 and variance 1. Consider the time series . The autocovariance function is

$$g(k)\; =\; z^k.$$

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 periods.

When

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. When lies in , is the Fourier transform of .

See also

• Bochner-Minlos theorem
• Characteristic function (probability theory)
• Positive-definite function on a group

References

• • M. Reed and Barry Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.