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In [[mathematics]], the term '''positive-definite function''' may refer to a couple of different concepts.
==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|publisher=Springer|year=1967}}</ref> This definition is in clear conflict with the one below.▼
== Most common usage ==
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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).
▲==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|publisher=Springer|year=1967}}</ref> This definition is in
==References==
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