Positive-definite function

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

Most common usageEdit

A positive-definite function of a real variable x is a complex-valued function   such that for any real numbers x1, …, xn 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

 

(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).

ExamplesEdit

Bochner's theoremEdit

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.[1]

ApplicationsEdit

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).

GeneralizationEdit

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 definitionEdit

The following definition conflict 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  .[2][3] In physics, the requirement that   may be dropped (see, e.g., Corney and Olsen[4]).

See alsoEdit

ReferencesEdit

  • Christian Berg, Christensen, Paul Ressel. Harmonic Analysis on Semigroups, GTM, Springer Verlag.
  • Z. Sasvári, Positive Definite and Definitizable Functions, Akademie Verlag, 1994
  • Wells, J. H.; Williams, L. R. Embeddings and extensions in analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.

NotesEdit

  1. ^ Bochner, Salomon (1959). Lectures on Fourier integrals. Princeton University Press.
  2. ^ Verhulst, Ferdinand (1996). Nonlinear Differential Equations and Dynamical Systems (2nd ed.). Springer. ISBN 3-540-60934-2.
  3. ^ Hahn, Wolfgang (1967). Stability of Motion. Springer.
  4. ^ Corney, J. F.; Olsen, M. K. (19 February 2015). "Non-Gaussian pure states and positive Wigner functions". Physical Review A. 91 (2): 023824. arXiv:1412.4868. Bibcode:2015PhRvA..91b3824C. doi:10.1103/PhysRevA.91.023824. ISSN 1050-2947,1094-1622 Check |issn= value (help). S2CID 119293595.

External linksEdit