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In [[mathematics]], the term '''positive-definite function''' may refer to a couple of different concepts.
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==In dynamical systems==
 
A [[real number|real]]-valued, continuously differentiable [[function (mathematics)|function]] ''f'' is '''positive definite''' on a neighborhood of the origin, ''D'', if <math>f(0)=0</math> and <math>f(x)>0</math> for every non-zero <math>x\in D</math>.<ref>{{cite book|last=Verhulst|first=Ferdinand|title=Nonlinear Differential Equations and Dynamical Systems|edition=2nd ed.|publisher=Springer|year=1996|isbn=3-540-60934-2}}</ref><ref>{{cite book|last=Hahn|first=Wolfgang|title=Stability of Motion|publisher=Springer|year=1967}}</ref>
 
A function is '''negative definite''' if the inequality is reversed. A function is '''semidefinite''' if the strong inequality is replaced with a weak (<math> \geq\,</math> or <math> \leq\,</math>) one.
 
==In analysis==
 
A '''positive-definite function''' of a real variable ''x'' is a [[complex number|complex]]-valued function ''f'':'''R''' &rarr; '''C''' such that for any real numbers ''x''<sub>1</sub>, ..., ''x''<sub>n</sub> the ''n''&times;''n'' [[matrix (mathematics)|matrix]]
 
:<math> A = (a_{i,j})_{i,j=1}^n~, \quad a_{ij} = f(x_i - x_j) </math>
 
is [[positive-definite matrix|positive '''semi-'''definite]] (which requires ''A'' to be [[Hermitian matrix|Hermitian]]; therefore ''f''(-''x'') is the [[complex conjugate]] of ''f''(''x'')).
 
In particular, it is necessary (but not sufficient) that
 
:<math> f(0) \geq 0~, \quad |f(x)| \leq f(0) </math>
 
(these inequalities follow from the condition for ''n''=1,2.)
 
===Bochner's theorem===
{{main|Bochner's theorem}}
 
Positive-definiteness arises naturally in the theory of the [[Fourier transform]]; it is easy to see 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'') &ge; 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 (mathematics)|measure]].<ref>{{cite book | last=Bochner | first=Salomon | authorlink=Salomon Bochner | title=Lectures on Fourier integrals | publisher=Princeton University Press | year=1959}}</ref>
 
====Applications====
 
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 <math>R^d</math> 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 this context, one does not usually use Fourier terminology and instead one states that ''f(x)'' is the [[characteristic function (probability theory)|characteristic function]] of a [[symmetric]] [[probability density function|PDF]].
 
===Generalisation===
{{main|Positive-definite function on a group}}
 
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 space]]s (i.e. the theory of [[unitary representation]]s).
 
==References==
* 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.
 
==Notes==
<references/>
 
==External links==
* {{springer|title=Positive-definite function|id=p/p073890}}
 
[[Category:Complex analysis]]
[[Category:Dynamical systems]]
[[Category:Types of functions]]

Latest revision as of 20:53, 11 March 2014

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