In mathematics, the term positive-definite function may refer to a couple of different concepts.
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A real-valued, continuously differentiable function f is positive definite on a neighborhood of the origin, D, if and for every non-zero .[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 ( or ) one.
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 (in particular, A should 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.)
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.[3]
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 and one requires that points that are closely separated have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an n-by-n matrix) is always positive definite. One strategy is to define a correlation matrix A which is then multiplied by a scalar to give a covariance matrix: this must be positive definite. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f()), then function f() must be positive definite to ensure the covariance matrix A is positive definite. See Kriging.
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.
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).