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

Contents

[edit] 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, which can be shown via a direct calculation.

[edit] 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. [1]

In fact the map

t \; \stackrel{\Phi}{\mapsto} \; U_t

is a strongly continuous representation of the additive group R. [2]

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) 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.

[edit] Notes

  1. ^ It is sufficient to verify the unitarity of [f] for f in F0(R), since such functions are dense in \mathcal{H}. If en is the element in F0(R) defned by en(m) = δnm, then
    \langle U_t [e_n], U_t [e_m] \rangle = \langle[e_{n+t}], [e_{m+t}]\rangle = K( m+t , n+t) = K(m,n) = Q_{n-m}.\;
  2. ^ We have
    \langle U_t [e_x], [e_x] \rangle = \langle U_t [e_x], [e_x] \rangle = K(x-t , x) = Q(t).
    Since Q is continuous, Φ is weakly continuous. Since the unitaries lies in the unit ball, Φ is also strongly continuous.

[edit] Reference

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