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== Most common usage ==
A '''positive-definite function''' of a real variable ''x'' is a [[complex number|complex]]-valued function <math> f: \mathbb{R} \mapsto \mathbb{C} </math> such that for any real numbers ''x''<sub>1</sub>, …, ''x''<sub>n</sub> the ''n''×''n'' [[matrix (mathematics)|matrix]]
:<math> A = \left(a_{i,j}\right)_{i,j=1}^n~, \quad a_{i,j} = 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
(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).
===Examples===
{{main|Bochner's theorem}}
Positive-definiteness arises naturally in the theory of the [[Fourier transform]]; it iscan easybe to seeseen 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 (mathematics)|measure]].<ref>{{cite book | last=Bochner | first=Salomon | authorlink=Salomon Bochner | title=Lectures on Fourier integrals | url=https://archive.org/details/lecturesonfourie0000boch | url-access=registration | 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> andare onetaken and requires that points that are mutually close 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'' to be positive-definite. See [[Kriging]].
In this context, one doesFourier notterminology usuallyis usenot Fouriernormally terminologyused and instead oneit is statesstated that ''f(x)'' is the [[characteristic function (probability theory)|characteristic function]] of a [[symmetric]] [[probability density function|probability density function (PDF)]].
===Generalization===
==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|publisher=Springer|year=1996|isbn=3-540-60934-2}}</ref><ref>{{cite book|last=Hahn|first=Wolfgang|title=Stability of Motion|url=https://archive.org/details/stabilityofmotio0000hahn|url-access=registration|publisher=Springer|year=1967}}</ref> This definition is in conflict with the one above.
==See also==
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