- 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 thefunctional 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 function s 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
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