Английская Википедия:Bochner's theorem
Шаблон:Use American English Шаблон:Short description Шаблон:About 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.)[1]
The theorem for locally compact abelian groups
Bochner's theorem for a locally compact abelian group G, with dual group <math>\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>\widehat{G}</math> such that
- <math>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>\widehat{G}</math>. Conversely, the Fourier transform of a probability measure on <math>\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>(\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" Ug defined by (Ug)(h) (g') = h(gШаблон:' − g), for a representative of [h], is unitary. So the map
- <math>g \mapsto U_g</math>
is a unitary representations of G on <math>(\mathcal{H}, \langle \cdot, \cdot\rangle_f)</math>. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have
- <math>\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>\langle \cdot [e], [e] \rangle_f </math> on C*(G) is the pull-back of a state on <math>C_0(\widehat{G})</math>, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives
- <math>\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>\widehat{G}</math>, the function
- <math>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>C_0(\widehat{G})</math>. This extends uniquely to a representation of its multiplier algebra <math>C_b(\widehat{G})</math> and therefore a strongly continuous unitary representation Ug. As above we have f given by some vector state on Ug
- <math>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>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>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>\{f_n\}</math> of mean 0 is a (wide-sense) stationary time series if the covariance
- <math>\operatorname{Cov}(f_n, f_m)</math>
only depends on n − m. The function
- <math>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>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>\mathbb{Z}</math>. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that
- <math>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>\{z^n f\}</math>. The autocovariance function is
- <math>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>\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
- Шаблон:Citation
- M. Reed and Barry Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
- Шаблон:Citation
- Английская Википедия
- Theorems in harmonic analysis
- Theorems in measure theory
- Theorems in functional analysis
- Theorems in Fourier analysis
- Theorems in statistics
- Страницы, где используется шаблон "Навигационная таблица/Телепорт"
- Страницы с телепортом
- Википедия
- Статья из Википедии
- Статья из Английской Википедии