A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak ( or ) one.
A positive-definite function of a real variable x is a complex-valued function
such that for any real numbers
the n×n matrix A with entries
If a function f is positive semidefinite, we find by taking n = 1 that
By taking n=2 and recognising that a positive-definite matrix has a positive determinant we get
Positive-definiteness 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.
This result generalizes 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).
In statistics, the theorem is usually applied to real functions; the matrix A is then multiplied by a scalar to give a covariance matrix, which must be positive definite. In a statistical context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function of a symmetric PDF.