In mathematics, the term positive-definite function may refer to a couple of different concepts.
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In dynamical systems
A real-valued, continuously differentiable function f is positive definite on a neighborhood of the origin, D, if
Most common usage
A positive-definite function of a real variable x is a complex-valued function f:R → C such that for any real numbers x1, ..., xn the n×n matrix
is positive semi-definite (which requires A to be Hermitian; therefore f(-x) is the complex conjugate of f(x)).
In particular, it is necessary (but not sufficient) that
(these inequalities follow from the condition for n=1,2.)
A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak (≤,≥0).
Bochner's theorem
Positive-definiteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite it is sufficient for f 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 any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.
Applications
In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Typically, one takes n scalar measurements of some scalar value at points in
In this context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function of a symmetric PDF.
Generalization
One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).