In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
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).
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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.
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 mutually close 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.
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).
In dynamical systems
- Christian Berg, Christensen, Paul Ressel. Harmonic Analysis on Semigroups, GTM, Springer Verlag.
- Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, 1994
- Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.