Content deleted Content added
use latex instead plain text |
Bluelink 2 books for verifiability.) #IABot (v2.0) (GreenC bot |
||
Line 24:
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'') ≥ 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) [[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====
Line 39:
==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|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==
| |||