66.215.184.32: /* Quasiperiodic signals */ Changed how the fractions are displayed.

2025-06-06T22:22:40Z

Quasiperiodic signals: Changed how the fractions are displayed.

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{{Short description|Class of functions behaving "like" periodic functions}}
{{distinguish|Almost periodic function|Quasi-periodic oscillation}}
{{More citations needed|date=January 2023}}
In [[mathematics]], a '''quasiperiodic function''' is a [[function (mathematics)|function]] that has a certain similarity to a [[periodic function]].<ref>{{Cite book |last=Mitropolsky |first=Yu A. |url=https://www.worldcat.org/oclc/840309575 |title=Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients |date=1993 |publisher=Springer Netherlands |others=A. M. Samoilenko, D. I. Martinyuk |isbn=978-94-011-2728-8 |location=Dordrecht |pages=108 |language=en |oclc=840309575}}</ref> A function <math>f</math> is quasiperiodic with quasiperiod <math>\omega</math> if <math>f(z + \omega) = g(z,f(z))</math>, where <math>g</math> is a "''simpler''" function than <math>f</math>. What it means to be "''simpler''" is vague.

[[File:Arithmetic quasiperiodic function.gif|thumb|350px|The function ''f''(''x'') = {{sfrac|''x''|2π}} + sin(''x'') satisfies the equation ''f''(''x''+2π) = ''f''(''x'') + 1, and is hence arithmetic quasiperiodic.]]
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:

:<math>f(z + \omega) = f(z) + C</math>

Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:

:<math>f(z + \omega) = C f(z)</math>

An example of this is the [[theta function|Jacobi theta function]], where

:<math>\vartheta(z+\tau;\tau) = e^{-2\pi iz - \pi i\tau}\vartheta(z;\tau),</math>

shows that for fixed <math>\tau</math> it has quasiperiod <math>\tau</math>; it also is periodic with period one. Another example is provided by the [[Weierstrass sigma function]], which is quasiperiodic in two independent quasiperiods, the periods of the corresponding [[Weierstrass elliptic functions|Weierstrass ''℘'' function]]. [[Bloch's theorem]] says that the eigenfunctions of a periodic Schrödinger equation (or other periodic linear equations) can be found in quasiperiodic form, and a related form of quasi-periodic solution for periodic linear differential equations is expressed by [[Floquet theory]].

Functions with an additive functional equation

:<math> f(z + \omega) = f(z)+az+b \ </math>
are also called quasiperiodic. An example of this is the [[Weierstrass zeta function]], where

:<math> \zeta(z + \omega, \Lambda) = \zeta(z , \Lambda) + \eta (\omega , \Lambda) \ </math>

for a ''z''-independent η when ω is a period of the corresponding Weierstrass ℘ function.

In the special case where <math>f(z + \omega)=f(z)</math> we say ''f'' is [[periodic function|periodic]] with period ω in the period lattice <math>\Lambda</math>.

==Quasiperiodic signals==

Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions in the sense defined here; instead they have the nature of [[almost periodic function]]s and that article should be consulted. The more vague and general notion of [[quasiperiodicity]] has even less to do with quasiperiodic functions in the mathematical sense.

A useful example is the function:

:<math>f(z) = \sin(Az) + \sin(Bz)</math>

If the ratio {{fraction|A|B}} is [[rational number|rational]], this will have a true period, but if {{fraction|A|B}} is [[irrational number|irrational]] there is no true period, but a succession of increasingly accurate "almost" periods.

== See also ==
* [[Quasiperiodic motion]]

==References==
{{Reflist}}

==External links==
*[https://web.archive.org/web/20070713223414/http://planetmath.org/encyclopedia/QuasiperiodicFunction.html Quasiperiodic function] at [[PlanetMath]]

[[Category:Complex analysis]]
[[Category:Types of functions]]

Quasiperiodic function - Revision history

66.215.184.32: /* Quasiperiodic signals */ Changed how the fractions are displayed.