Bochner's Theorem
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In mathematics, Bochner's theorem (named for
Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Au ...
) characterizes the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of a positive finite Borel measure on the real line. More generally in
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
, Bochner's theorem asserts that under Fourier transform a continuous
positive-definite function In mathematics, a positive-definite function is, depending on the context, either of two types of function. Most common usage A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb suc ...
on a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
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 \widehat, 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 In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
''μ'' on \widehat such that : f(g) = \int_ \xi(g) \,d\mu(\xi), i.e. ''f'' is the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of a unique probability measure ''μ'' on \widehat. Conversely, the Fourier transform of a probability measure on \widehat 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between the group
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
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 In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: * the final topology on the disjoint union * the top ...
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ' ...
s 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''0(''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''1, ''g''2) = ''f''(''g''1 − ''g''2) induces a (possibly degenerate)
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on ''F''0(''G''). Quotiening out degeneracy and taking the completion gives a Hilbert space : (\mathcal, \langle \cdot, \cdot\rangle_f), whose typical element is an equivalence class 'h'' For a fixed ''g'' in ''G'', the "
shift operator In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...
" ''Ug'' defined by (''Ug'')(''h'') (g') = ''h''(''g'' − ''g''), for a representative of 'h'' is unitary. So the map : g \mapsto U_g is a unitary representations of ''G'' on (\mathcal, \langle \cdot, \cdot\rangle_f). By continuity of ''f'', it is weakly continuous, therefore strongly continuous. By construction, we have : \langle U_g \rangle_f = 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 \cdot \rangle_f on C*(''G'') is the pull-back of a state on C_0(\widehat), which is necessarily integration against a probability measure ''μ''. Chasing through the isomorphisms then gives : \langle U_g \rangle_f = \int_ \xi(g) \,d\mu(\xi). On the other hand, given a probability measure ''μ'' on \widehat, the function : f(g) = \int_ \xi(g) \,d\mu(\xi) is a normalized continuous positive-definite function. Continuity of ''f'' follows from the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
. For positive-definiteness, take a nondegenerate representation of C_0(\widehat). This extends uniquely to a representation of its multiplier algebra C_b(\widehat) and therefore a strongly continuous unitary representation ''Ug''. As above we have ''f'' given by some vector state on ''Ug'' : f(g) = \langle U_g v, v \rangle, therefore positive-definite. The two constructions are mutual inverses.


Special cases

Bochner's theorem in the special case of the
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
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) = \int_ e^ \,d\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) = \int_ e^ \,d\mu(\xi).


Applications

In statistics, Bochner's theorem can be used to describe the
serial correlation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
of certain type of
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. E ...
. A sequence of random variables \ of mean 0 is a (wide-sense) stationary time series if the
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
: \operatorname(f_n, f_m) only depends on ''n'' − ''m''. The function : g(n - m) = \operatorname(f_n, f_m) is called the autocovariance function of the time series. By the mean zero assumption, : g(n - m) = \langle f_n, f_m \rangle, where ⟨⋅, ⋅⟩ denotes the inner product on the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
of random variables with finite second moments. It is then immediate that ''g'' is a positive-definite function on the integers \mathbb. By Bochner's theorem, there exists a unique positive measure ''μ'' on , 1such that : g(k) = \int e^ \,d\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'' ∈ , 1 and ''f'' be a random variable of mean 0 and variance 1. Consider the time series \. The autocovariance function is : g(k) = z^k. 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 In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
with respect to the Lebesgue measure, and its Radon–Nikodym derivative ''f'' is called the
spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of the time series. When ''g'' lies in \ell^1(\mathbb), ''f'' is the Fourier transform of ''g''.


See also

* Positive-definite function on a group *
Characteristic function (probability theory) In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is ...


References

* * M. Reed and Barry Simon, ''Methods of Modern Mathematical Physics'', vol. II, Academic Press, 1975. * {{Functional analysis Theorems in harmonic analysis Theorems in measure theory Theorems in functional analysis Theorems in Fourier analysis Theorems in statistics