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

Theorem For any normalized continuous positive-definite function ${\displaystyle f}$  on ${\displaystyle G}$  (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 (see Herglotz representation 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

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

Similarly, a continuous function ${\displaystyle f}$  on ${\displaystyle \mathbb {R} }$  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

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

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

• 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