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.

Background

Given a positive finite Borel measure μ on the real line R, the Fourier transform Q of μ is the continuous function

Q(t)=\int _{\mathbb {R} }e^{-itx}d\mu (x).

Q is continuous since for a fixed x, the function e^-itx is continuous and periodic. The function Q is a positive definite function, i.e. the kernel K(x, y) = Q(y - x) is positive definite; this can be checked via a direct calculation.

The theorem

Bochner's theorem says the converse is true, i.e. every positive definite function Q is the Fourier transform of a positive finite Borel measure. A proof can be sketched as follows.

Let F₀(R) be the family of complex valued functions on R with finite support, i.e. f(x) = 0 for all but finitely many x. The positive definite kernel K(x, y) induces a sesquilinear form on F₀(R). This in turn results in a Hilbert space

({\mathcal {H}},\langle \;,\;\rangle )

whose typical element is an equivalence class [g]. For a fixed t in R, the "shift operator" U_t defined by (U_tg)(x) = g(x - t), for a representative of [g] is unitary. In fact the map

t\;{\stackrel {\Phi }{\mapsto }}\;U_{t}

is a strongly continuous representation of the additive group R. By Stone's theorem, there exists a (possibly unbounded) self-adjoint operator A such that

U_{-t}=e^{-iAt}.\;

This implies there exists a finite positive Borel measure μ on R where

\langle U_{-t}[e_{0}],[e_{0}]\rangle =\int e^{-iAt}d\mu (x),

where e₀ is the element in F₀(R) defined by e₀(m) = 1 if m = 0 and 0 otherwise. Because

\langle U_{-t}[e_{0}],[e_{0}]\rangle =K(-t,0)=Q(t),

the theorem holds.

The theorem for locally compact abelian groups

If G is a locally compact Abelian group with dual group ${\widehat {G}}$ , then any normalized positive definite function f on G is the Fourier transform of a probability measure μ on ${\widehat {G}}$ , so that

f(g)=\int _{\widehat {G}}\xi (g)d\mu (\xi ).

In fact continuous unitary representations π of G with a cyclic unit vector v correspond to continuous positive definite functions f on G through the Gelfand–Naimark construction

f(g)=(\pi (g)v,v).

Each such representation correponds to a continuous non-degenerate *-representation of the convolution algebra L¹(G) and hence, by Fourier transform, its C* algebra $C_{0}({\widehat {G}})$ . Matrix coefficients of such *-representations correspond to probability measures on ${\widehat {G}}$ .

Applications

In statistics, one often has to specify a covariance matrix, the rows and columns of which correspond to observations of some phenomenon. The observations are made at points $x_{i},i=1,\ldots ,n$ in some space. This matrix is to be a function of the positions of the observations and one usually insists that points which are close to one another have high covariance. One usually specifies that the covariance matrix $\Sigma =\sigma ^{2}A$ where $\sigma ^{2}$ is a scalar and matrix $A$ is n by n with ones down the main diagonal. Element $i,j$ of $A$ (corresponding to the correlation between observation i and observation j) is then required to be $f\left(x_{i}-x_{j}\right)$ for some function $f(\cdot )$ , and because $A$ must be positive definite, $f(\cdot )$ must be a positive definite function. Bochner's theorem shows that $f(.)$ must be the characteristic function of a symmetric PDF.