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).
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen 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, n scalar measurements of some scalar value at points in are taken and points that are mutually close are required to have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an n × 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).
The following definition conflicts with the one above.
In dynamical systems, a real-valued, continuously differentiable function f can be called positive-definite on a neighborhood D of the origin if and for every non-zero . In physics, the requirement that may be dropped (see, e.g., Corney and Olsen).
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