Positive-definite function

In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage.
A "positive-definite function" of a real variable "x" is a complex-valued function formula_1 such that for any real numbers "x"1, …, "x""n" 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
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 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.
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, "n" scalar measurements of some scalar value at points in formula_4 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 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, Fourier terminology is not normally used and instead it is stated that "f"("x") is the characteristic function of a symmetric probability density function (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).
Alternative definition.
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 formula_5 and formula_6 for every non-zero formula_7. In physics, the requirement that formula_5 may be dropped (see, e.g., Corney and Olsen).