Positive-definite function: Difference between revisions

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{{Short description|Bimodal function}}

In [[mathematics]], a '''positive-definite function''' is, depending on the context, either of two types of [[function (mathematics)|function]].

== MostDefinition common usage1 ==

ALet ''positive-definite<math>\mathbb{R}</math> function''be the set of a [[real number|real]]s variableand ''x''<math>\mathbb{C}</math> isbe athe set of [[complex number|complex]]-valueds.

A function <math> f: \mathbb{R} \to \mathbb{C} </math> such thatis called ''positive semi-definite'' if for anyall real numbers ''x''₁, …, ''x''_''n'' the ''n''&thinsp;×&thinsp;''n'' [[matrix (mathematics)|matrix]]

:<math> A = \left(a_{ij}\right)_{i,j=1}^n~, \quad a_{ij} = f(x_i - x_j) </math>

is a [[positive-definite matrix|positive ''semi-''definite]] (which requires ''A'' to be [[Hermitian matrix|Hermitian]];.{{citation thereforeneeded|date=June ''f''(−''x'') is the [[complex conjugate]] of ''f''(''x'')).2023}}

By definition, a positive semi-definite matrix, such as <math>A</math>, is [[Hermitian matrix|Hermitian]]; therefore ''f''(−''x'') is the [[complex conjugate]] of ''f''(''x'')).

In particular, it is necessary (but not sufficient) that

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u^* A^{(g_y)} u

= \sum_{j, k = 1}^{n} \overline{u_k} u_j e^{i \langle y, x_k - x_j \rangle}

= \sum_{k = 1}^{n} \overline{u_k} u_j e^{i \langle y, x_k \rangle} \sum_{j = 1}^{n} u_j e^{- i \langle y, x_j \rangle}

= \left| \sum_{j = 1}^{n} \overline{u_j} e^{i \langle y, x_j \rangle} \right|^2

\ge 0.

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\ge 0,

</math>

where <math>\tilde{A}^{(f)} = \big( f(\phi(x_i) - \phi(x_j)) = f(\tilde{x}_i - \tilde{x}_j) \big)_{i, j}</math> where <math>\tilde{x}_k := \phi(x_k)</math> are distinct as <math>\phi</math> is [[linear]].<ref>{{cite book |last1=Cheney |first1=Elliot Ward |title=A course in Approximation Theory |date=2009 |publisher=American Mathematical Society |isbn=9780821847985 |pages=77–78 |url=https://books.google.com/books?id=II6DAwAAQBAJ |access-date=3 February 2022}}</ref>

===Bochner's theorem===

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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).

==Definition 2==

==Alternative definition==

Alternatively, a function <math>f : \reals^n \to \reals</math> is 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>

TheNote followingthat this definition conflicts with thedefinition 1, onegiven above.

In dynamical systems, a [[real number|real]]-valued, [[continuously differentiable function|continuously differentiable]] function ''f'' can 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> In physics, the requirement that <math>f(0) = 0</math> mayis besometimes 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 |pages=023824|volume=91|issue=2|doi=10.1103/PhysRevA.91.023824|arxiv=1412.4868|bibcode=2015PhRvA..91b3824C|s2cid=119293595}}</ref>).

==See also==

* [[Positive definiteness]]

* [[Positive-definite kernel]]