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.

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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" Ut defined by (Utg)(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

\displaystyle{f(g)=(\pi(g)v,v).}

Each such representation correponds to a continuous non-degenerate *-representation of the convolution algebra L1(G)

\Pi(\varphi)=\int_G \varphi(g) \pi(g)\, dg

and hence, by Fourier transform, of its C* algebra C_0(\widehat{G}).

On the other hand matrix coefficients of non-degenerate continuous *-representations of C0(X) with X a locally compact space, in this case \widehat{G}, correspond to probability measures on X.

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^2A 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.

See also

References