Quasiperiodic function

In mathematics, a function "f" is said to be quasiperiodic with "quasiperiod" (sometimes simply called the "period") ω if for certain constants "a" and "b", "f" satisfies the functional equation

:$f(z\; +\; omega)\; =\; exp(az+b)\; f(z).$

An example of this is the Jacobi theta function, where

:$vartheta(z+\; au;\; au)\; =\; exp(-2\; pi\; i\; z\; -\; pi\; i\; au)vartheta(z;\; au),$

shows that for fixed τ it has quasiperiod τ; 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 ℘ function.

Functions with an additive functional equation

:$f(z\; +\; omega)\; =\; f(z)+az+b$are also called quasiperiodic. An example of this is the Weierstrass zeta function, where

:$zeta(z\; +\; omega)\; =\; zeta(z)\; +\; eta$

for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function.

In the special case where $f(z\; +\; omega)=f(z)$ we say "f" is periodic with period ω.

Quasiperiodic signals

Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions; instead they have the nature of almost periodic functions and that article should be consulted.

See also

* Quasiperiodic motion
* Almost periodic function

External links

* [http://planetmath.org/encyclopedia/QuasiperiodicFunction.html Quasiperiodic function] at PlanetMath