Bochner's theorem for a locally compact abelian groupG, with dual group, says the following: Theorem For any normalized continuous positive-definite function f on G, there exists a unique probability measureμ on such that i.e. f is the Fourier transform of a unique probability measure μ on. Conversely, the Fourier transform of a probability measure on 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* and C0. 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. 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 be the family of complex-valued functions on G with finite support, i.e. h = 0 for all but finitely manyg. The positive-definite kernelK = f induces a inner product on F0. Quotiening out degeneracy and taking the completion gives a Hilbert space whose typical element is an equivalence class . For a fixed g in G, the "shift operator" Ug defined by = h, for a representative of , is unitary. So the map is a unitary representations of G on. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have where 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 on C* is the pull-back of a state on, 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, the function is a normalized continuous positive-definite function. Continuity of ffollows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of. This extends uniquely to a representation of its multiplier algebra and therefore a strongly continuous unitary representation Ug. As above we have f given by some vector state on Ug therefore positive-definite. The two constructions are mutual inverses.
Special cases
Bochner's theorem in the special case of the discrete groupZ is often referred to as Herglotz's theorem and says that a function f on Z with f = 1 is positive-definite if and only if there exists a probability measure μ on the circleT such that Similarly, a continuous functionf on R with f = 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 of mean 0 is a 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 ℤ. By Bochner's theorem, there exists a unique positive measure μ on 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 and f be a random variable of mean 0 and variance 1. Consider the time series. 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 continuouswith respect to the Lebesgue measure, and its Radon–Nikodym derivativef is called the spectral density of the time series. When g lies in ℓ1, f is the Fourier transform of g.