Bochner's theorem

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

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 "F"₀(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 "F"₀(R). This in turn results in a Hilbert space

:$(\; mathcal\{H\},\; langle\; ;,;\; angle\; )$

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

:$langle\; U\_\{-t\}\; [e\_0]\; ,\; [e\_0]\; angle\; =\; int\; e^\{-iAt\}\; d\; mu(x)\; ,$

where "e"₀ is the element in "F"₀(R) defined by "e"₀("m") = 1 if "m" = 0 and 0 otherwise. Because

:$langle\; U\_\{-t\}\; [e\_0]\; ,\; [e\_0]\; angle\; =\; K(-t,0)\; =\; Q(t),$

the theorem holds.

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 $fleft(x\_i-x\_j\; ight)$ 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

References

* M. Reed and B. Simon, "Methods of Modern Mathematical Physics", vol. II, Academic Press, 1975.