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| 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 group|positive definite function]] on a [[locally compact group|locally compact abelian group]] corresponds to a finite positive measure on the [[Pontryagin duality|Pontryagin dual group]].
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| ==The theorem for locally compact abelian groups==
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| Bochner's theorem for a locally compact Abelian group ''G'', with dual group <math>\widehat{G}</math>, says the following:
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| '''Theorem''' For any normalized continuous positive definite function ''f'' on ''G'' (normalization here means ''f'' is 1 at the unit of ''G''), there exists a unique [[probability measure]] on <math>\widehat{G}</math> such that
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| :<math> f(g)=\int_{\widehat{G}} \xi(g) d\mu(\xi),</math> | | .mwe-math-fallback-image-inline, .mwe-math-fallback-image-display { |
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| 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 <math>\widehat{G}</math> is necessarily a normalized continuous positive definite function ''f'' on ''G''. This is in fact a one-to-one correspondence.
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| The [[Fourier transform#Gelfand transform|Gelfand-Fourier transform]] is an [[isomorphism]] between the group [[C*-algebra]] C*(''G'') and C<sub>0</sub>(''G''^). The theorem is essentially the dual statement for [[state (functional analysis)|state]]s of the two Abelian C*-algebras.
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| The proof of the theorem passes through vector states on [[strong operator topology|strongly continuous]] [[unitary representation]]s of ''G'' (the proof in fact shows every normalized continuous positive definite function must be of this form).
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| 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''<sub>0</sub>(''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''<sub>1</sub>, ''g''<sub>2</sub>) = ''f''(''g''<sub>1</sub> - ''g''<sub>2</sub>) induces a (possibly degenerate) [[inner product]] on ''F''<sub>0</sub>(''G''). Quotiening out degeneracy and taking the completion gives a Hilbert space
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| :<math>( \mathcal{H}, \langle \;,\; \rangle_f )</math> | |
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| whose typical element is an equivalence class [''h'']. For a fixed ''g'' in ''G'', the "[[shift operator]]" ''U<sub>g</sub>'' defined by (''U<sub>g</sub>'')('' h '') (g') = ''h''(''g' - g''), for a representative of [''h''], is unitary. So the map
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| :<math>g \; \mapsto \; U_g</math> | |
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| is a unitary representations of ''G'' on <math>( \mathcal{H}, \langle \;,\; \rangle_f )</math>. By continuity of ''f'', it is weakly continuous, therefore strongly continuous. By construction, we have
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| :<math>\langle U_{g} [e], [e] \rangle_f = f(g)</math> | |
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| 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
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| :<math>\langle U_{g} [e], [e] \rangle_f = \int_{\widehat{G}} \xi(g) d\mu(\xi).</math>
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| On the other hand, given a probability measure μ on <math>\widehat{G}</math>, the function
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| :<math>f(g) = \int_{\widehat{G}} \xi(g) d\mu(\xi).</math>
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| 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 ''U<sub>g</sub>''. As above we have ''f'' given by some vector state on ''U<sub>g</sub>''
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| :<math>f(g) = \langle U_g v, v \rangle,</math>
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| therefore positive-definite.
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| The two constructions are mutual inverses.
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| == Special cases ==
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| 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
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| :<math>f(k) = \int_{\mathbb{T}} e^{-2 \pi i k x}d \mu(x).</math>
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| 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
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| :<math>f(t) = \int_{\mathbb{R}} e^{-2 \pi i \xi t} d \mu(\xi).</math>
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| ==Applications==
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| 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 stochastic process|stationary time series]] if the [[covariance]]
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| :<math>\mbox{Cov}(f_n, f_m)</math>
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| only depends on ''n''-''m''. The function
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| :<math>g(n-m) = \mbox{Cov}(f_n, f_m)</math>
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| is called the [[autocovariance function]] of the time series. By the mean zero assumption,
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| :<math>g(n-m) = \langle f_n, f_m \rangle</math>
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| where ⟨⋅ , ⋅⟩ denotes the inner product on the [[Hilbert space]] of random variables with finite second moments. It is then immediate that
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| ''g'' is a positive definite function on the integers ℤ. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that
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| :<math>g(k) = \int e^{-2 \pi i k x} d \mu(x)</math>.
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| This measure μ is called the '''spectral measure''' of the time series. It yields information about the "seasonal trends" of the series.
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| 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
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| :<math>g(k) = z^k</math>. | |
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| 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.
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| 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 ''l''<sup>1</sup>(ℤ), ''f'' is the Fourier transform of ''g''.
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| == See also ==
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| * [[Positive definite function on a group]]
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| * [[Characteristic function (probability theory)]]
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| ==References==
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| *{{citation|last=Loomis|first= L. H.|title=An introduction to abstract harmonic analysis|publisher= Van Nostrand|year= 1953}}
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| * M. Reed and B. Simon, ''Methods of Modern Mathematical Physics'', vol. II, Academic Press, 1975.
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| *{{citation|last=Rudin|first= W.|title=Fourier analysis on groups|publisher=Wiley-Interscience|year= 1990|isbn= 0-471-52364-X}}
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| [[Category:Theorems in harmonic analysis]]
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| [[Category:Theorems in measure theory]]
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| [[Category:Theorems in functional analysis]]
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| [[Category:Theorems in Fourier analysis]]
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| [[Category:Statistical theorems]]
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