# 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.)

## The theorem for locally compact abelian groups

Bochner's theorem for a locally compact abelian group $G$, with dual group ${\widehat {G}}$, 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 $\mu$ on ${\widehat {G}}$ such that
$f(g)=\int _{\widehat {G}}\xi (g)\,d\mu (\xi ),$
i.e. $f$ is the Fourier transform of a unique probability measure $\mu$ on ${\widehat {G}}$. Conversely, the Fourier transform of a probability measure on ${\widehat {G}}$ 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 $C_{0}({\widehat {G}})$. 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 $G$ (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 on $F_{0}(G)$. Quotienting out degeneracy and taking the completion gives a Hilbert space
$({\mathcal {H}},\langle \cdot ,\cdot \rangle _{f}),$
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
$g\mapsto U_{g}$
is a unitary representations of $G$ on $({\mathcal {H}},\langle \cdot ,\cdot \rangle _{f})$. By continuity of $f$, it is weakly continuous, therefore strongly continuous. By construction, we have
$\langle U_{g}[e],[e]\rangle _{f}=f(g),$
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 $\langle \cdot [e],[e]\rangle _{f}$ on $C^{*}(G)$ is the pullback of a state on $C_{0}({\widehat {G}})$, which is necessarily integration against a probability measure $\mu$. Chasing through the isomorphisms then gives
$\langle U_{g}[e],[e]\rangle _{f}=\int _{\widehat {G}}\xi (g)\,d\mu (\xi ).$
On the other hand, given a probability measure $\mu$ on ${\widehat {G}}$, the function
$f(g)=\int _{\widehat {G}}\xi (g)\,d\mu (\xi )$
is a normalized continuous positive-definite function. Continuity of $f$ follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of $C_{0}({\widehat {G}})$. This extends uniquely to a representation of its multiplier algebra $C_{b}({\widehat {G}})$ and therefore a strongly continuous unitary representation $U_{g}$. As above we have $f$ given by some vector state on $U_{g}$
$f(g)=\langle U_{g}v,v\rangle ,$
therefore positive-definite.
The two constructions are mutual inverses.

## Special cases

Bochner's theorem in the special case of the discrete group $\mathbb {Z}$ is often referred to as Herglotz's theorem (see Herglotz representation theorem) and says that a function $f$ on $\mathbb {Z}$ with $f(0)=1$ is positive-definite if and only if there exists a probability measure $\mu$ on the circle $\mathbb {T}$ such that
$f(k)=\int _{\mathbb {T} }e^{-2\pi ikx}\,d\mu (x).$
Similarly, a continuous function $f$ on $\mathbb {R}$ with $f(0)=1$ is positive-definite if and only if there exists a probability measure $\mu$ on $\mathbb {R}$ such that
$f(t)=\int _{\mathbb {R} }e^{-2\pi i\xi t}\,d\mu (\xi ).$

## 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 $\{f_{n}\}$ of mean 0 is a (wide-sense) stationary time series if the covariance
$\operatorname {Cov} (f_{n},f_{m})$
only depends on $n-m$. The function
$g(n-m)=\operatorname {Cov} (f_{n},f_{m})$
is called the autocovariance function of the time series. By the mean zero assumption,
$g(n-m)=\langle f_{n},f_{m}\rangle ,$
where $\langle \cdot ,\cdot \rangle$ 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 $\mathbb {Z}$. By Bochner's theorem, there exists a unique positive measure $\mu$ on $[0,1]$ such that
$g(k)=\int e^{-2\pi ikx}\,d\mu (x).$
This measure $\mu$ 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 $1/m\in [0,1]$) and $f$ be a random variable of mean 0 and variance 1. Consider the time series $\{z^{n}f\}$. 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 $\mu$ 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 $\ell ^{1}(\mathbb {Z} )$, $f$ is the Fourier transform of $g$.