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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 &mu; on the real line R, the Fourier transform Q of &mu; 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 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 &mu; 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) defned 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.

Bochner's theorem can be generalized. Instead of positive definite function Q, one can consider distributions of positive type. Bochner-Schwarz theorem then states that a distribution is of positive type if and only if it is a tempered distribution and the Fourier transform of a positive measure of at most polynomial growth.

Reference

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