Bochner's theorem · Lablypedia

The theorem for locally compact abelian groups

Bochner's theorem for a locally compact abelian group $G$ , with dual group $\hat{G}$ , says the following:

Theorem For any normalized continuous positive-definite function $f : G \to C$ (normalization here means that $f$ is 1 at the unit of $G$ ), there exists a unique probability measure $μ$ on $\hat{G}$ such that

$f (g) = \int_{\hat{G}} ξ (g) d μ (ξ),$

i.e. $f$ is the Fourier transform of a unique probability measure $μ$ on $\hat{G}$ . Conversely, the Fourier transform of a probability measure on $\hat{G}$ 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 $C_{0} (\hat{G})$ . 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 that 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 $F_{0} (G)$ . Quotienting out degeneracy and taking the completion gives a Hilbert space

$(H, ⟨ \cdot, \cdot ⟩_{f}),$

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 $(H, ⟨ \cdot, \cdot ⟩_{f})$ . By continuity of $f$ , it is weakly continuous, therefore strongly continuous. By construction, we have

$⟨ U_{g} [e], [e] ⟩_{f} = f (g),$

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 $⟨ \cdot [e], [e] ⟩_{f}$ on $C^{*} (G)$ is the pullback of a state on $C_{0} (\hat{G})$ , which is necessarily integration against a probability measure $μ$ . Chasing through the isomorphisms then gives

$⟨ U_{g} [e], [e] ⟩_{f} = \int_{\hat{G}} ξ (g) d μ (ξ) .$

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

$f (g) = \int_{\hat{G}} ξ (g) d μ (ξ)$

is a normalized continuous positive-definite function. Continuity of $f$ follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of $C_{0} (\hat{G})$ . This extends uniquely to a representation of its multiplier algebra $C_{b} (\hat{G})$ and therefore a strongly continuous unitary representation $U_{g}$ . As above we have $f$ given by some vector state on $U_{g}$

$f (g) = ⟨ U_{g} v, v ⟩,$

therefore positive-definite.

The two constructions are mutual inverses.

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Special cases

Bochner's theorem in the special case of the discrete group $Z$ is often referred to as Herglotz's 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 $f (k) = \int_{T} e^{- 2 π i k x} d μ (x),$ are the coefficients of a Fourier-Stieltjes series.

Similarly, a continuous function $f : R^{d} \to C$ with $f (0) = 1$ is positive-definite if and only if there exists a probability measure $μ$ on $R^{d}$ such that

$f (t) = \int_{R^{d}} e^{- 2 π i ξ \cdot t} d μ (ξ) .$

Here, $f$ is positive definite if for any finite set of points $α_{1}, \dots, α_{N} \in R^{d}$ , and any complex numbers $ρ_{1}, \dots, ρ_{N} \in C$ , there holds

$\sum_{p, q = 1}^{N} f (α_{p} - α_{q}) ρ_{p} {\bar{ρ}}_{q} ⩾ 0.$

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

$Cov (f_{n}, f_{m})$

only depends on $n - m$ . The function

$g (n - m) = Cov (f_{n}, f_{m})$

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

$g (n - m) = ⟨ f_{n}, f_{m} ⟩,$

where $⟨ \cdot, \cdot ⟩$ 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

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

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 $1 / m \in [0, 1]$ ) 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 $ℓ^{1} (Z)$ , $f$ is the Fourier transform of $g$ .

The theorem for locally compact abelian groups

Special cases

Applications

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