positive-definite function

In mathematics, the term positive-definite function may refer to a couple of different concepts.

Gegenbauer polynomials also appear in the theory of Positive-definite functions.

Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.

Conversely, every operator-valued positive-definite function arises in this way.

Thus, can be defined as the linear span of the set of continuous positive-definite functions on .

Positive-definite functions in classical analysis.

This condition applies for example to potentials that are: a) positive functions; b) positive-definite functions.

One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context.

The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.

On the other hand, consider now a positive-definite function F on G. A unitary representation of G can be obtained as follows.

Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).

In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups.

In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions.

Positive-definite functions on G are intimately related to unitary representations of G. Every unitary representation of G gives rise to a family of positive-definite functions.

A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions defined by a symmetric, positive-definite function called the reproducing kernel such that the function belongs to for all .