Bochner's theorem

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 : G \to \mathbb{C} (normalization here means that f is 1 at the unit of G), there exists a unique probability measure \mu 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 \mu 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 C_0(\widehat{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 (\mathcal{H}, \langle \cdot, \cdot\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 \cdot, \cdot\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 pullback of a state on C_0(\widehat{G}), which is necessarily integration against a probability measure \mu. 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 \mu 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 \mathbb{Z} is often referred to as Herglotz's theorem and says that a function f on \mathbb{Z} with f(0) = 1 is positive-definite if and only if there exists a probability measure \mu on the circle \mathbb{T} such that f(k) = \int_{\mathbb{T}} e^{-2 \pi i k x} \,d\mu(x), are the coefficients of a Fourier-Stieltjes series. Similarly, a continuous function f : \mathbb{R}^d \to \mathbb{C} with f(0) = 1 is positive-definite if and only if there exists a probability measure \mu on \mathbb{R}^d such that f(t) = \int_{\mathbb{R}^d} e^{-2 \pi i \xi \cdot t} \,d\mu(\xi). Here, f is positive definite if for any finite set of points \alpha_1, \cdots, \alpha_N \in \mathbb{R}^d, and any complex numbers \rho_1, \cdots, \rho_N \in \mathbb{C}, there holds \sum_{p,q = 1}^N f(\alpha_p - \alpha_q) \rho_p \bar{\rho}_q \geqslant 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 \operatorname{Cov}(f_n, f_m) only depends on n - m. The function g(n - m) = \operatorname{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 \langle\cdot, \cdot\rangle 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 \mathbb{Z}. By Bochner's theorem, there exists a unique positive measure \mu on [0, 1] such that g(k) = \int e^{-2 \pi i k x} \,d\mu(x). This measure \mu 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 \mu 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 \ell^1(\mathbb{Z}), f is the Fourier transform of g. == See also ==