Bochner's theorem

Bochner's theorem for a locally compact abelian group $G$, with dual group $\backslash widehat\{G\}$, 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 $\backslash mu$ on $\backslash widehat\{G\}$ such that

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

i.e. $f$ is the Fourier transform of a unique probability measure $\backslash mu$ on $\backslash widehat\{G\}$. Conversely, the Fourier transform of a probability measure on $\backslash widehat\{G\}$ 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 $C\_0(\backslash widehat\{G\})$. 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

$$(\backslash mathcal\{H\},\; \backslash langle\; \backslash cdot,\; \backslash cdot\backslash rangle\_f),$$

whose typical element is an equivalence class $[h]$. For a fixed $g$ in $G$, the "shift operator" $U\_g$ defined by $(U\_g\; h)\; (g\text{'})\; =\; h(g\text{'}\; -\; g)$, for a representative of $[h]$, is unitary. So the map

$$g\; \backslash mapsto\; U\_g$$

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

$$\backslash langle\; U\_g\; [e],\; [e]\; \backslash rangle\_f\; =\; f(g),$$

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 $\backslash langle\; \backslash cdot\; [e],\; [e]\; \backslash rangle\_f$ on $C^*(G)$ is the pullback of a state on $C\_0(\backslash widehat\{G\})$, which is necessarily integration against a probability measure $\backslash mu$. Chasing through the isomorphisms then gives

$$\backslash langle\; U\_g\; [e],\; [e]\; \backslash rangle\_f\; =\; \backslash int\_\{\backslash widehat\{G\}\}\; \backslash xi(g)\; \backslash ,d\backslash mu(\backslash xi).$$

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

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

is a normalized continuous positive-definite function. Continuity of $f$ follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of $C\_0(\backslash widehat\{G\})$. This extends uniquely to a representation of its multiplier algebra $C\_b(\backslash widehat\{G\})$ and therefore a strongly continuous unitary representation $U\_g$. As above we have $f$ given by some vector state on $U\_g$

$$f(g)\; =\; \backslash langle\; U\_g\; v,\; v\; \backslash rangle,$$

therefore positive-definite.

The two constructions are mutual inverses.

Bochner's theorem in the special case of the discrete group $\backslash mathbb\{Z\}$ is often referred to as Herglotz's theorem (see Herglotz representation theorem) and says that a function $f$ on $\backslash mathbb\{Z\}$ with $f(0)\; =\; 1$ is positive-definite if and only if there exists a probability measure $\backslash mu$ on the circle $\backslash mathbb\{T\}$ such that

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

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

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

In statistics, Bochner's theorem can be used to describe the serial correlation of certain type of time series. A sequence of random variables $\backslash \{f\_n\backslash \}$ of mean 0 is a (wide-sense) stationary time series if the covariance

$$\backslash operatorname\{Cov\}(f\_n,\; f\_m)$$

only depends on $n\; -\; m$. The function

$$g(n\; -\; m)\; =\; \backslash operatorname\{Cov\}(f\_n,\; f\_m)$$

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

$$g(n\; -\; m)\; =\; \backslash langle\; f\_n,\; f\_m\; \backslash rangle,$$

where $\backslash langle\backslash cdot,\; \backslash cdot\backslash rangle$ 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 $\backslash mathbb\{Z\}$. By Bochner's theorem, there exists a unique positive measure $\backslash mu$ on $[0,\; 1]$ such that

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

This measure $\backslash mu$ 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 $1/m\; \backslash in\; [0,\; 1]$) and $f$ be a random variable of mean 0 and variance 1. Consider the time series $\backslash \{z^n\; f\backslash \}$. 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 $\backslash mu$ 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 $\backslash ell^1(\backslash mathbb\{Z\})$, $f$ is the Fourier transform of $g$.

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

== Notes ==

{{Reflist}}

• {{citation|last=Loomis|first= L. H.|title=An introduction to abstract harmonic analysis|publisher= Van Nostrand|year= 1953}}
• M. Reed and Barry Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975.
• {{citation|last=Rudin|first= W.|title=Fourier analysis on groups|publisher=Wiley-Interscience|year= 1990|isbn= 0-471-52364-X}}

{{Functional analysis}}

Category:Theorems in harmonic analysis

Category:Theorems in measure theory

Category:Theorems in functional analysis

Category:Theorems in Fourier analysis

Category:Theorems in statistics