Positive-definite function - Definition and Overview

In mathematics, a positive-definite function of a real variable x is a function

f:R → C

such that for any real numbers

x₁, ...,x_n

the n×n matrix A with entries

a_ij = f(x_i − x_j)

is positive semi-definite. It is usual to restrict to the case in which f(−x) is the complex conjugate of f(x), making the matrix A Hermitian.

For example, taking n = 1 we must have

f(0) ≥ 0

and taking n = 2 the product

f(x − y)f(y − x) ≤ f(0)²;

therefore necessarily

|f(x)| ≤ f(0).

This condition arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite is a necessary condition on f, for it to be the Fourier transform of a function g on the real line with g(y) ≥ 0.

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

This result generalises to the context of Pontryagin duality, with positive-definite functions defined on any locally compact abelian topological group. Positive-definite functions also occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).