Positive-definite function: Difference between revisions

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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).

==Alternative definition==

==In dynamical systems==

The following definition conflict with the one above.

A [[real number|real]]-valued, [[continuously differentiable function|continuously differentiable]] function ''f'' is ''positive-definite'' on a [[neighborhood (mathematics)|neighborhood]] ''D'' of the origin 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.

AIn dynamical systems, a [[real number|real]]-valued, [[continuously differentiable function|continuously differentiable]] function ''f'' iscan be called ''positive-definite'' on a [[neighborhood (mathematics)|neighborhood]] ''D'' of the origin 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> ThisIn definitionphysics, isthe inrequirement conflictthat with<math>f(0) the= one0</math> abovemay be dropped (see, e.g., Corney and Olsen<ref>{{cite journal|first1=J. F.|last1=Corney|first2=M. K.|last2=Olsen|title=Non-Gaussian pure states and positive Wigner functions|url=http://arxiv.org/abs/1412.4868|journal=Physical Review A|date=19 February 2015|issn=1050-2947,1094-1622|pages=023824|volume=91|issue=2|doi=10.1103/PhysRevA.91.023824}}</ref>).

==See also==