Позитивті анықталған функциялар және Бохнер теоремасы

Bimodal function
In mathematics, a positive definite function is, depending on the context, either of two types of function.

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.

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. 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).

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).

Alternatively, a function is called positive definite on a neighborhood D of the origin if and for every non zero

Note that this definition conflicts with definition 1, given above. In physics, the requirement that is sometimes dropped (see, e. g., Corney and Olsen).