Bochner's theorem

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In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line.

1 Background
2 The theorem
3 Applications
4 See also
5 References

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.

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 the Stone-von Neumann 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.

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 $Σ = σ 2 A$ where $σ 2$ 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.