Bochner's theorem

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group.

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" 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 Stone's 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.

The theorem for locally compact abelian groups

Bochner's theorem for a locally compact Abelian group G, with dual group ${\widehat {G}}$, says the following:

Theorem For any normalized continuous positive definite function f on G (normalization here means f is 1 at the unit of G), there exists a unique probability measure on ${\widehat {G}}$ such that

$f(g)=\int _{\widehat {G}}\xi (g)d\mu (\xi ),$

i.e. f is the Fourier transform of a unique probability measure μ on ${\widehat {G}}$. Conversely, the Fourier transform of a probability measure ${\widehat {G}}$ is necessarily a normalized continuous positive definite function f on G. This is in fact a one-to-one correspondence.

The Gelfand-Fourier transform is an isomorphism between the group C*-algebra C*(G) and C₀(G^). The theorem is essentially the dual statement for states of the two Abelian C*-algebras.

The proof of the theorem passes through vector states on strongly continuous unitary representations of G (the proof in fact shows every normalized continuous positive definite function must be of this form).

Given a normalized continuous positive definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F₀(G) be the family of complex valued functions on G with finite support, i.e. h(g) = 0 for all but finitely many g. The positive definite kernel K(g₁, g₂) = f(g₁ - g₂) induces a (possibly degenerate) inner product on F₀(G). Quotiening out degeneracy and taking the completion gives a Hilbert space

$({\mathcal {H}},\langle \;,\;\rangle _{f})$

whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" U_g defined by (U_g)( h ) (g') = h(g' - g), for a representative of [h], is unitary. So the map

$g\;\mapsto \;U_{g}$

is a unitary representations of G on $({\mathcal {H}},\langle \;,\;\rangle _{f})$. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have

$\langle U_{g}[e],[e]\rangle _{f}=f(g)$

where [e] is the class of the function that is 1 on the identity of G and zero elsewhere. But by Gelfand-Fourier isomorphism, the vector state $\langle \cdot [e],[e]\rangle _{f}$ on C*(G) is the pull-back of a state on $C_{0}({\widehat {G}})$, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

$\langle U_{g}[e],[e]\rangle _{f}=\int _{\widehat {G}}\xi (g)d\mu (\xi ).$

On the other hand, given a probability measure μ on ${\widehat {G}}$, the function

$f(g)=\int _{\widehat {G}}\xi (g)d\mu (\xi ).$

is a normalized continuous positive definite function. Continuity of f follows from the dominated convergence theorem. For positive definitness, take a nondegenerate representation of $C_{0}({\widehat {G}})$. This extends uniquely to a representation of its multiplier algebra $C_{b}({\widehat {G}})$ and therefore a strongly continuous unitary representation U_g. As above we have f given by some vector state on U_g

$f(g)=\langle U_{g}v,v\rangle ,$

therefore positive-definite.

The two constructions are mutual inverses.

Special cases

For the discrete group Z, Bochner's theorem says a function f on Z with f(0) = 1 is positive definite if and only if there exists a unique probability measure μ on the circle T such that

$f(k)=\int _{\mathbb {T} }e^{-2\pi ikx}d\mu (x).$

Similarly, a continuous function f on R with f(0) = 1 is positive definite if and only if there exists a unique probability measure μ on R such that

$f(t)=\int _{\mathbb {R} }e^{-2\pi i\xi t}d\mu (\xi ).$

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 $\Sigma =\sigma ^{2}A$ 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 $f\left(x_{i}-x_{j}\right)$ 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.

See also

• Positive definite function on a group
• Characteristic function (probability theory)

References

• Loomis, L. H. (1953), An introduction to abstract harmonic analysis, Van Nostrand
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
• Rudin, W. (1990), Fourier analysis on groups, Wiley-Interscience, ISBN 0-471-52364-X