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{{Short description|Bimodal function}}
==
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 <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]]▼
▲A
:<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'')).▼
is a [[positive-definite matrix|positive ''semi-''definite matrix]].{{citation needed|date=June 2023}}
▲
In particular, it is necessary (but not sufficient) that
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(these inequalities follow from the condition for ''n'' = 1, 2.)
A function is ''negative semi-definite'' if the inequality is reversed. A function is ''
===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
:<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===
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Positive-definiteness arises naturally in the theory of the [[Fourier transform]]; it can be seen 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 function|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.
In this context, Fourier terminology is not normally used and instead it is stated that ''f''(''x
===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==
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>).
▲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==
* [[Positive definiteness]]
* [[Positive-definite kernel]]
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