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

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

Each such representation corresponds 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 Σ = σ²A where σ² 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

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

References

• Loomis, L. H. (1953), An introduction to abstract harmonic analysis, Van Nostrand 
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
• Rudin, W. (1990), Fourier analysis on groups, Wiley-Interscience, ISBN 0-471-52364-X