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 case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.)
Bochner's theorem for a locally compact abelian group G, with dual group
\widehat{G}
Theorem For any normalized continuous positive-definite function f on G (normalization here means that f is 1 at the unit of G), there exists a unique probability measure μ on
\widehat{G}
f(g)=\int\widehat{G
i.e. f is the Fourier transform of a unique probability measure μ on
\widehat{G}
\widehat{G}
The Gelfand–Fourier transform is an isomorphism between the group C*-algebra C*(G) and C0(Ĝ). 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 that 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
(l{H},\langle ⋅ , ⋅ \ranglef),
whose typical element is an equivalence class [''h'']. For a fixed g in G, the "shift operator" Ug defined by (Ug)(h) (g') = h(g − g), for a representative of [''h''], is unitary. So the map
g\mapstoUg
is a unitary representations of G on
(l{H},\langle ⋅ , ⋅ \ranglef)
\langleUg[e],[e]\ranglef=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 ⋅ [e],[e]\ranglef
C0(\widehat{G})
\langleUg[e],[e]\ranglef=\int\widehat{G
On the other hand, given a probability measure μ on
\widehat{G}
f(g)=\int\widehat{G
is a normalized continuous positive-definite function. Continuity of f follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of
C0(\widehat{G})
Cb(\widehat{G})
f(g)=\langleUgv,v\rangle,
therefore positive-definite.
The two constructions are mutual inverses.
Bochner's theorem in the special case of the discrete group Z is often referred to as Herglotz's theorem (see Herglotz representation theorem) and says that a function f on Z with f(0) = 1 is positive-definite if and only if there exists a probability measure μ on the circle T such that
f(k)=\intTe-2d\mu(x).
Similarly, a continuous function f on R with f(0) = 1 is positive-definite if and only if there exists a probability measure μ on R such that
f(t)=\intRe-2d\mu(\xi).
In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables
\{fn\}
\operatorname{Cov}(fn,fm)
only depends on n − m. The function
g(n-m)=\operatorname{Cov}(fn,fm)
is called the autocovariance function of the time series. By the mean zero assumption,
g(n-m)=\langlefn,fm\rangle,
where ⟨⋅, ⋅⟩ denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that g is a positive-definite function on the integers
Z
g(k)=\inte-2d\mu(x).
This measure μ is called the spectral measure of the time series. It yields information about the "seasonal trends" of the series.
For example, let z be an m-th root of unity (with the current identification, this is 1/m ∈ [0, 1]) and f be a random variable of mean 0 and variance 1. Consider the time series
\{znf\}
g(k)=zk.
Evidently, the corresponding spectral measure is the Dirac point mass centered at z. This is related to the fact that the time series repeats itself every m periods.
When g has sufficiently fast decay, the measure μ is absolutely continuous with respect to the Lebesgue measure, and its Radon–Nikodym derivative f is called the spectral density of the time series. When g lies in
\ell1(Z)