Content deleted Content added
→Applications: correct |
No edit summary |
||
| (26 intermediate revisions by 17 users not shown) | |||
Line 1:
{{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
Line 15 ⟶ 19:
(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===
Line 25 ⟶ 51:
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. Typically, ''n'' scalar measurements of some scalar value at points in <math>R^d</math> are taken and 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''
In this context, Fourier terminology is not normally used and instead it is stated that ''f''(''x
===Generalization===
Line 38 ⟶ 64:
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]]
| |||