Bochner's theorem

Short description: Theorem of Fourier transforms of Borel measures

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.)

The theorem for locally compact abelian groups

Bochner's theorem for a locally compact abelian group G, with dual group [math]\displaystyle{ \widehat{G} }[/math], says the following:

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 [math]\displaystyle{ \widehat{G} }[/math] such that

[math]\displaystyle{ f(g) = \int_{\widehat{G}} \xi(g) \,d\mu(\xi), }[/math]

i.e. f is the Fourier transform of a unique probability measure μ on [math]\displaystyle{ \widehat{G} }[/math]. Conversely, the Fourier transform of a probability measure on [math]\displaystyle{ \widehat{G} }[/math] 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(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

[math]\displaystyle{ (\mathcal{H}, \langle \cdot, \cdot\rangle_f), }[/math]

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

[math]\displaystyle{ g \mapsto U_g }[/math]

is a unitary representations of G on [math]\displaystyle{ (\mathcal{H}, \langle \cdot, \cdot\rangle_f) }[/math]. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have

[math]\displaystyle{ \langle U_g [e], [e] \rangle_f = f(g), }[/math]

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 [math]\displaystyle{ \langle \cdot [e], [e] \rangle_f }[/math] on C*(G) is the pull-back of a state on [math]\displaystyle{ C_0(\widehat{G}) }[/math], which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

[math]\displaystyle{ \langle U_g [e], [e] \rangle_f = \int_{\widehat{G}} \xi(g) \,d\mu(\xi). }[/math]

On the other hand, given a probability measure μ on [math]\displaystyle{ \widehat{G} }[/math], the function

[math]\displaystyle{ f(g) = \int_{\widehat{G}} \xi(g) \,d\mu(\xi) }[/math]

is a normalized continuous positive-definite function. Continuity of f follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of [math]\displaystyle{ C_0(\widehat{G}) }[/math]. This extends uniquely to a representation of its multiplier algebra [math]\displaystyle{ C_b(\widehat{G}) }[/math] and therefore a strongly continuous unitary representation U_g. As above we have f given by some vector state on U_g

[math]\displaystyle{ f(g) = \langle U_g v, v \rangle, }[/math]

therefore positive-definite.

The two constructions are mutual inverses.

Special cases

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

[math]\displaystyle{ f(k) = \int_{\mathbb{T}} e^{-2 \pi i k x} \,d\mu(x). }[/math]

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

[math]\displaystyle{ f(t) = \int_{\mathbb{R}} e^{-2 \pi i \xi t} \,d\mu(\xi). }[/math]

Applications

In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables [math]\displaystyle{ \{f_n\} }[/math] of mean 0 is a (wide-sense) stationary time series if the covariance

[math]\displaystyle{ \operatorname{Cov}(f_n, f_m) }[/math]

only depends on n − m. The function

[math]\displaystyle{ g(n - m) = \operatorname{Cov}(f_n, f_m) }[/math]

is called the autocovariance function of the time series. By the mean zero assumption,

[math]\displaystyle{ g(n - m) = \langle f_n, f_m \rangle, }[/math]

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 [math]\displaystyle{ \mathbb{Z} }[/math]. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that

[math]\displaystyle{ g(k) = \int e^{-2 \pi i k x} \,d\mu(x). }[/math]

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 [math]\displaystyle{ \{z^n f\} }[/math]. The autocovariance function is

[math]\displaystyle{ g(k) = z^k. }[/math]

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 [math]\displaystyle{ \ell^1(\mathbb{Z}) }[/math], f is the Fourier transform of g.

See also

• Bochner-Minlos theorem
• Characteristic function (probability theory)
• Positive-definite function on a group

References

• Loomis, L. H. (1953), An introduction to abstract harmonic analysis, Van Nostrand 
• M. Reed and Barry 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