Bochner's theorem

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 theorem for locally compact abelian groups

Bochner's theorem for a locally compact abelian group G, with dual group $\widehat{G}$, 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 $\widehat{G}$ such that

$f(g)=\int_{\widehat{G}} \xi(g) d\mu(\xi),$

i.e. f is the Fourier transform of a unique probability measure μ on $\widehat{G}$. Conversely, the Fourier transform of a probability measure on $\widehat{G}$ 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 C₀(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 F₀(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(g₁, g₂) = f(g₁ - g₂) induces a (possibly degenerate) inner product on F₀(G). Quotiening out degeneracy and taking the completion gives a Hilbert space

$( \mathcal{H}, \langle \;,\; \rangle_f )$

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

$g \; \mapsto \; U_g$

is a unitary representations of G on $( \mathcal{H}, \langle \;,\; \rangle_f )$. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have

$\langle U_{g} [e], [e] \rangle_f = f(g)$

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 $\langle \cdot [e], [e] \rangle_f$ on C*(G) is the pull-back of a state on $C_0(\widehat{G})$, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

$\langle U_{g} [e], [e] \rangle_f = \int_{\widehat{G}} \xi(g) d\mu(\xi).$

On the other hand, given a probability measure μ on $\widehat{G}$, the function

$f(g) = \int_{\widehat{G}} \xi(g) d\mu(\xi).$

is a normalized continuous positive definite function. Continuity of f follows from the dominated convergence theorem. For positive definiteness, take a nondegenerate representation of $C_0(\widehat{G})$. This extends uniquely to a representation of its multiplier algebra $C_b(\widehat{G})$ and therefore a strongly continuous unitary representation U_g. As above we have f given by some vector state on U_g

$f(g) = \langle U_g v, v \rangle,$

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

$f(k) = \int_{\mathbb{T}} e^{-2 \pi i k x}d \mu(x).$

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

$f(t) = \int_{\mathbb{R}} e^{-2 \pi i \xi t} d \mu(\xi).$

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 $\{ f_n \}$ of mean 0 is a (wide-sense) stationary time series if the covariance

$\mbox{Cov}(f_n, f_m)$

only depends on n-m. The function

$g(n-m) = \mbox{Cov}(f_n, f_m)$

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

$g(n-m) = \langle f_n, f_m \rangle$

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

$g(k) = \int e^{-2 \pi i k x} d \mu(x)$.

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 $\{ z^n f \}$. The autocovariance function is

$g(k) = z^k$.

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 l¹(ℤ), f is the Fourier transform of g.

See also

• Positive definite function on a group
• Characteristic function (probability theory)

References

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