Positive-definite_function

Positive-definite function

In mathematics, the term positive-definite function may refer to a couple of different concepts.

In dynamical systems

A real-valued, continuously differentiable function f is positive definite on a neighborhood of the origin, D, if f(0)=0 and f(x)>0 for every non-zero xin D.

A function is negative definite if the inequality is reversed. A function is semidefinite if the strong inequality is replaced with a weak (geq or leq) one.

In complex analysis and statistics

A positive-definite function of a real variable x is a complex-valued function

f:RC

such that for any real numbers

x1, ..., xn

the n×n matrix A with entries

aij = f(xixj)

is a positive semi-definite matrix. It is usual to restrict to the case in which f(−x) is the complex conjugate of f(x), making the matrix A Hermitian.

If a function f is positive semidefinite, we find by taking n = 1 that

f(0) ≥ 0.

By taking n=2 and recognising that a positive-definite matrix has a positive determinant we get

f(xy)f(yx) ≤ f(0)2

which implies

|f(x)| ≤ f(0).

Positive-definiteness arises naturally in the theory of the Fourier transform; it is easy to see directly that to be positive-definite is a necessary condition on f, for it 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 a continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.

This result generalizes to the context of Pontryagin duality, with positive-definite functions defined on any locally compact abelian topological group. Positive-definite functions also occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).

In statistics, the theorem is usually applied to real functions; the matrix A is then multiplied by a scalar to give a covariance matrix, which must be positive definite. In a statistical context, one does not usually use Fourier terminology and instead one states that f(x) is the characteristic function of a symmetric PDF.

See also

References

  • Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, 1994

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