Bochner's theorem

In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier-Stieltjes transform of a positive finite Borel measure on the real line.^[1] 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^[2][3]

Bochner's theorem for a locally compact abelian group ${\displaystyle G}$, with dual group ${\displaystyle {\widehat {G}}}$, says the following:^[4]

Theorem For any normalized continuous positive-definite function ${\displaystyle f:G\to \mathbb {C} }$ (normalization here means that ${\displaystyle f}$ is 1 at the unit of ${\displaystyle G}$), there exists a unique probability measure ${\displaystyle \mu }$ on ${\displaystyle {\widehat {G}}}$ such that

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

i.e. ${\displaystyle f}$ is the Fourier transform of a unique probability measure ${\displaystyle \mu }$ on ${\displaystyle {\widehat {G}}}$. Conversely, the Fourier transform of a probability measure on ${\displaystyle {\widehat {G}}}$ is necessarily a normalized continuous positive-definite function ${\displaystyle f}$ on ${\displaystyle G}$. This is in fact a one-to-one correspondence.

The Gelfand–Fourier transform is an isomorphism between the group C*-algebra ${\displaystyle C^{*}(G)}$ and ${\displaystyle C_{0}({\widehat {G}})}$. 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 ${\displaystyle G}$ (the proof in fact shows that every normalized continuous positive-definite function must be of this form).

Given a normalized continuous positive-definite function ${\displaystyle f}$ on ${\displaystyle G}$, one can construct a strongly continuous unitary representation of ${\displaystyle G}$ in a natural way: Let ${\displaystyle F_{0}(G)}$ be the family of complex-valued functions on ${\displaystyle G}$ with finite support, i.e. ${\displaystyle h(g)=0}$ for all but finitely many ${\displaystyle g}$. The positive-definite kernel ${\displaystyle K(g_{1},g_{2})=f(g_{1}-g_{2})}$ induces a (possibly degenerate) inner product on ${\displaystyle F_{0}(G)}$. Quotienting 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 ${\displaystyle [h]}$. For a fixed ${\displaystyle g}$ in ${\displaystyle G}$, the "shift operator" ${\displaystyle U_{g}}$ defined by ${\displaystyle (U_{g}h)(g')=h(g'-g)}$, for a representative of ${\displaystyle [h]}$, is unitary. So the map

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

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

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

where ${\displaystyle [e]}$ is the class of the function that is 1 on the identity of ${\displaystyle G}$ and zero elsewhere. But by Gelfand–Fourier isomorphism, the vector state ${\displaystyle \langle \cdot [e],[e]\rangle _{f}}$ on ${\displaystyle C^{*}(G)}$ is the pullback of a state on ${\displaystyle C_{0}({\widehat {G}})}$, which is necessarily integration against a probability measure ${\displaystyle \mu }$. 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 ${\displaystyle \mu }$ 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 ${\displaystyle f}$ 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 ${\displaystyle U_{g}}$. As above we have ${\displaystyle f}$ given by some vector state on ${\displaystyle U_{g}}$

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

therefore positive-definite.

The two constructions are mutual inverses.

Bochner's theorem in the special case of the discrete group ${\displaystyle \mathbb {Z} }$ is often referred to as Herglotz's theorem and says that a function ${\displaystyle f}$ on ${\displaystyle \mathbb {Z} }$ with ${\displaystyle f(0)=1}$ is positive-definite if and only if there exists a probability measure ${\displaystyle \mu }$ on the circle ${\displaystyle \mathbb {T} }$ such that $${\displaystyle f(k)=\int _{\mathbb {T} }e^{-2\pi ikx}\,d\mu (x),}$$ are the coefficients of a Fourier-Stieltjes series.^[5][6]

Similarly, a continuous function ${\displaystyle f:\mathbb {R} ^{d}\to \mathbb {C} }$ with ${\displaystyle f(0)=1}$ is positive-definite if and only if there exists a probability measure ${\displaystyle \mu }$ on ${\displaystyle \mathbb {R} }$ such that

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

Here, ${\displaystyle f}$ is positive definite if for any finite set of points ${\displaystyle \alpha _{1},\cdots ,\alpha _{N}\in \mathbb {R} ^{d}}$, and any complex numbers ${\displaystyle \rho _{1},\cdots ,\rho _{N}\in \mathbb {C} }$, there holds

$${\displaystyle \sum _{p,q=1}^{N}f(\alpha _{p}-\alpha _{q})\rho _{p}{\bar {\rho }}_{q}\geqslant 0.}$$

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 ${\displaystyle n-m}$. 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 ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the inner product on the Hilbert space of random variables with finite second moments. It is then immediate that ${\displaystyle g}$ is a positive-definite function on the integers ${\displaystyle \mathbb {Z} }$. By Bochner's theorem, there exists a unique positive measure ${\displaystyle \mu }$ on ${\displaystyle [0,1]}$ such that

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

This measure ${\displaystyle \mu }$ is called the spectral measure of the time series. It yields information about the "seasonal trends" of the series.

For example, let ${\displaystyle z}$ be an ${\displaystyle m}$-th root of unity (with the current identification, this is ${\displaystyle 1/m\in [0,1]}$) and ${\displaystyle f}$ 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 ${\displaystyle z}$. This is related to the fact that the time series repeats itself every ${\displaystyle m}$ periods.

When ${\displaystyle g}$ has sufficiently fast decay, the measure ${\displaystyle \mu }$ is absolutely continuous with respect to the Lebesgue measure, and its Radon–Nikodym derivative ${\displaystyle f}$ is called the spectral density of the time series. When ${\displaystyle g}$ lies in ${\displaystyle \ell ^{1}(\mathbb {Z} )}$, ${\displaystyle f}$ is the Fourier transform of ${\displaystyle g}$.

• Bochner-Minlos theorem
• Characteristic function (probability theory)
• Positive-definite function on a group
• Riesz–Markov–Kakutani representation theorem

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