Positive-definite function: Difference between revisions

Revision as of 15:05, 27 May 2018

Revision as of 12:15, 31 March 2019

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== Most common usage ==

A '''positive-definite function''' of a real variable ''x'' is a [[complex number|complex]]-valued function ''f'':'''R''' →→ '''C''' such that for any real numbers ''x''₁, ...…, ''x''_n the ''n''××''n'' [[matrix (mathematics)|matrix]]

:<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'')).

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 definite''' if the inequality is reversed. A function is '''semidefinite''' if the strong inequality is replaced with a weak (≤,≥0 ≥ 0).

===Examples===

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{{main|Bochner's theorem}}

Positive-definiteness arises naturally in the theory of the [[Fourier transform]]; it is easy to see 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 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 | publisher=Princeton University Press | year=1959}}</ref>

====Applications====

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==In dynamical systems==

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.

==See also==