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 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 ${\displaystyle {\widehat {G}}}$, 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 ${\displaystyle {\widehat {G}}}$ such that

${\displaystyle f(g)=\int _{\widehat {G}}\xi (g)\,d\mu (\xi ),}$

i.e. f is the Fourier transform of a unique probability measure μ on ${\displaystyle {\widehat {G}}}$. Conversely, the Fourier transform of a probability measure on ${\displaystyle {\widehat {G}}}$ 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

${\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle _{f}),}$

whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" U_g defined by (U_g)(h) (g') = h(g' − g), for a representative of [h], is unitary. So the map

${\displaystyle g\mapsto U_{g}}$

is a unitary representations of G on ${\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle _{f})}$. By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have

${\displaystyle \langle U_{g}[e],[e]\rangle _{f}=f(g),}$

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 ${\displaystyle \langle \cdot [e],[e]\rangle _{f}}$ on C*(G) is the pull-back of a state on ${\displaystyle C_{0}({\widehat {G}})}$, which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives

${\displaystyle \langle U_{g}[e],[e]\rangle _{f}=\int _{\widehat {G}}\xi (g)\,d\mu (\xi ).}$

On the other hand, given a probability measure μ on ${\displaystyle {\widehat {G}}}$, the function

${\displaystyle f(g)=\int _{\widehat {G}}\xi (g)\,d\mu (\xi )}$

is a normalized continuous positive-definite function. Continuity of f follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of ${\displaystyle C_{0}({\widehat {G}})}$. This extends uniquely to a representation of its multiplier algebra ${\displaystyle C_{b}({\widehat {G}})}$ and therefore a strongly continuous unitary representation U_g. As above we have f given by some vector state on U_g

${\displaystyle f(g)=\langle U_{g}v,v\rangle ,}$

therefore positive-definite.

The two constructions are mutual inverses.

Special cases

Bochner's theorem in the special case of the discrete group 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

${\displaystyle f(k)=\int _{\mathbb {T} }e^{-2\pi ikx}\,d\mu (x).}$

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

${\displaystyle f(t)=\int _{\mathbb {R} }e^{-2\pi i\xi t}\,d\mu (\xi ).}$

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 ${\displaystyle \{f_{n}\}}$ of mean 0 is a (wide-sense) stationary time series if the covariance

${\displaystyle \operatorname {Cov} (f_{n},f_{m})}$

only depends on n − m. The function

${\displaystyle g(n-m)=\operatorname {Cov} (f_{n},f_{m})}$

is called the autocovariance function of the time series. By the mean zero assumption,

${\displaystyle g(n-m)=\langle f_{n},f_{m}\rangle ,}$

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 ${\displaystyle \mathbb {Z} }$. By Bochner's theorem, there exists a unique positive measure μ on [0, 1] such that

${\displaystyle g(k)=\int e^{-2\pi ikx}\,d\mu (x).}$

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 ∈ [0, 1]) and f be a random variable of mean 0 and variance 1. Consider the time series ${\displaystyle \{z^{n}f\}}$. The autocovariance function is

${\displaystyle g(k)=z^{k}.}$

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 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 ${\displaystyle \ell ^{1}(\mathbb {Z} )}$, f is the Fourier transform of g.

See also

• Bochner-Minlos theorem
• Characteristic function (probability theory)
• Positive-definite function on a group

References

• Loomis, L. H. (1953), An introduction to abstract harmonic analysis, Van Nostrand
• M. Reed and Barry Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
• Rudin, W. (1990), Fourier analysis on groups, Wiley-Interscience, ISBN 0-471-52364-X