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 for locally compact abelian groups. Bochner's theorem for a locally compact abelian group "G", with dual group formula_1, 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 formula_1 such that i.e. "f" is the Fourier transform of a unique probability measure "μ" on formula_1. Conversely, the Fourier transform of a probability measure on formula_1 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 C("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 "F"("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"("g", "g") = "f"("g" − "g") induces a (possibly degenerate) inner product on "F"("G"). Quotiening out degeneracy and taking the completion gives a Hilbert  space whose typical element is an equivalence class ["h"]. For a fixed "g" in "G", the "shift operator" "U" defined by ("U")("h") (g') = "h"("g" − "g"), for a representative of ["h"], is unitary. So the map is a unitary representations of "G" on formula_8. By continuity of "f", it is weakly continuous, therefore strongly continuous. By construction, we have 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 formula_10 on C*("G") is the pull-back of a state on formula_11, which  is necessarily integration against a probability measure "μ". Chasing through the isomorphisms then gives On the other hand, given a probability measure "μ" on formula_1, the function is a normalized continuous positive-definite function. Continuity of "f" follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of formula_11. This extends uniquely to a representation of its multiplier algebra formula_16 and therefore a strongly continuous unitary representation "U". As above we have "f" given by some vector state on "U" 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 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 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 formula_20 of mean 0 is a (wide-sense) stationary time series if the covariance only depends on "n" − "m". The function is called the autocovariance function of the time series. By the mean zero assumption, 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 formula_24. By Bochner's theorem, there exists a unique positive measure "μ" on [0, 1] such that 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 formula_26. The autocovariance function is 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 formula_28, "f" is the Fourier transform of "g".