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 ==

Let <math>\mathbb{R}</math> be the set of [[real number]]s and <math>\mathbb{C}</math> be the set of [[complex number]]s.

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''₁, ..., ''x''_n the ''n''&times;''n'' [[matrix (mathematics)|matrix]]

A '''positive-definite function''' of<math> af: real\mathbb{R} variable\to ''x''\mathbb{C} is</math> a [[complexis number|complex]]-valued functioncalled ''f'':'''R'''positive &rarr; '''C'semi-definite'' such thatif for anyall real numbers ''x''₁, ...…, ''x''_''n'' the ''n''&timesthinsp;×&thinsp;''n'' [[matrix (mathematics)|matrix]]

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

:<math> A = \left(a_{i,jij}\right)_{i,j=1}^n~, \quad a_{i,jij} = 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'')).

is a [[positive-definite matrix|positive ''semi-''definite matrix]].{{citation needed|date=June 2023}}

isBy [[positive-definitedefinition, a matrix|positive '''semi-'''definite]] (whichmatrix, requiressuch ''as <math>A''</math>, to beis [[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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:<math> f(0) \geq 0~, \quad |f(x)| \leq f(0) </math>

(these inequalities follow from the condition for ''n'' = 1, 2.)

A function is '''negative semi-definite''' if the inequality is reversed. A function is '''semidefinite'definite'' if the strongweak inequality is replaced with a weakstrong (≤<,≥0 > 0).

===Examples===

If <math>(X, \langle \cdot, \cdot \rangle)</math> is a real [[inner product space]], then <math>g_y \colon X \to \mathbb{C}</math>, <math>x \mapsto \exp(i \langle y, x \rangle)</math> is positive definite for every <math>y \in X</math>: for all <math>u \in \mathbb{C}^n</math> and all <math>x_1, \ldots, x_n</math> we have

{{empty section|date=August 2017}}

:<math>

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} 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.

</math>

As nonnegative linear combinations of positive definite functions are again positive definite, the [[cosine function]] is positive definite as a nonnegative linear combination of the above functions:

:<math>

\cos(x) = \frac{1}{2} ( e^{i x} + e^{- i x}) = \frac{1}{2}(g_{1} + g_{-1}).

</math>

One can create a positive definite function <math>f \colon X \to \mathbb{C}</math> easily from positive definite function <math>f \colon \R \to \mathbb C</math> for any [[vector space]] <math>X</math>: choose a [[linear function]] <math>\phi \colon X \to \R</math> and define <math>f^* := f \circ \phi</math>.

Then

:<math>

u^* A^{(f^*)} u

= \sum_{j, k = 1}^{n} \overline{u_k} u_j f^*(x_k - x_j)

= \sum_{j, k = 1}^{n} \overline{u_k} u_j f(\phi(x_k) - \phi(x_j))

= u^* \tilde{A}^{(f)} u

\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===

{{main|Bochner's theorem}}

Positive-definiteness arises naturally in the theory of the [[Fourier transform]]; it iscan easy tobe 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'') &ge;≥ 0.

The converse result is '''[[Bochner's theorem]]''', stating that any [[continuous function|continuous]] positive-definite function on the real line is the Fourier transform of a (positive) [[Measuremeasure (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 requires thatand points that are mutually close are required to have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an {{nowrap|''n''-by-&thinsp;×&thinsp;''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 beis positive -definite. See [[Kriging]].

In this context, one doesFourier notterminology usuallyis usenot Fouriernormally terminologyused and instead oneit statesis stated that ''f''(''x)'') is the [[characteristic function (probability theory)|characteristic function]] of a [[symmetric]] [[probability density function|probability density function (PDF)]].

===Generalization===

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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==

==In dynamical systems==

AAlternatively, [[reala number|real]]-valued,function continuously<math>f differentiable: [[function\reals^n (mathematics)|function]]\to ''f''\reals</math> is 'called ''positive -definite''' on a [[neighborhood of the origin,(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.

Note that this definition conflicts with definition 1, given above.

In physics, the requirement that <math>f(0) = 0</math> is sometimes 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|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==

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 conflict with the one above.

* [[Positive definiteness]]

* [[Positive-definite kernel]]

==References==