Bochner's theorem

Bochner's theorem characterizes continuous positive-definite functions on locally compact abelian groups as Fourier transforms of unique finite positive measures on their dual groups.

Positive-definite function: A function whose matrices of pairwise values satisfy a nonnegative quadratic-form condition. This is the key property the theorem characterizes through Fourier transforms.
Positive linear functional: A linear functional that takes nonnegative values on nonnegative elements of its domain. Positivity lets the function define a positive functional from which its measure is recovered.
Bochner's theorem for positive-definite kernels: A representation theorem expressing translation-invariant positive-definite kernels through positive spectral measures. It converts stationary kernel structure into a distribution over frequencies.
Bochner's theorem on the real line: The characterization of continuous positive-definite functions on the real line as Fourier transforms of finite positive measures. This is the standard one-dimensional setting for the theorem.
Locally compact abelian group: A topological group that is locally compact and whose group operation is commutative. The theorem applies on precisely this class of groups.
Riesz representation theorem: A theorem representing positive linear functionals on suitable spaces of continuous functions by measures. It turns the positive functional associated with the function into a measure.
Mercer's theorem: A theorem expanding certain continuous positive kernels on compact spaces into eigenfunction series. It is a related spectral representation, using eigenfunctions rather than group characters.
Bochner's theorem on finite groups: The finite-group form of the theorem, relating positive-definite functions to nonnegative Fourier coefficients. Finite duality reduces the measure representation to a finite collection of nonnegative weights.
Pontryagin duality: A duality identifying locally compact abelian groups with groups of continuous characters on their duals. It supplies the dual group on which the representing measure lives.
Fourier–Stieltjes transform: The transform of a finite measure on a locally compact abelian group, evaluated on its characters. The theorem identifies the given function as exactly such a transform.

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