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.