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 .^[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 ( or ) one.

In complex analysis and statistics

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

f:R → C

such that for any real numbers

x₁, ..., x_n

the n×n matrix A with entries

a_ij = f(x_i − x_j)

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 semi-definite matrix has a nonnegative determinant we get

f(x − y)f(y − x) ≤ 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.^[3]

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

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 R^d and one requires that points that are closely separated 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.

In a this 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

• Positive definite function on a group

References

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

1. ^ Verhulst, Ferdinand (1996). Nonlinear Differential Equations and Dynamical Systems (2nd ed. ed.). Springer. ISBN 3-540-60934-2. 
2. ^ Hahn, Wolfgang (1967). Stability of Motion. Springer. 
3. ^ Bochner, Salomon (1959). Lectures on Fourier integrals. Princeton University Press.