Quasiperiodic function
From Wikipedia, the free encyclopedia
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
An example of this is the Jacobi theta function, where
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
are also called quasiperiodic. An example of this is the Weierstrass zeta function, where
- ζ(z + ω) = ζ(z) + η
for a fixed constant η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where f(z + ω) = f(z) we say f is periodic with period ω.
[edit] Generalized notion of quasiperiodicity
A function with a more general functional equation
- f(z + ω) = α(z)f(z)
can also be called quasiperiodic; this cannot be taken as an actual definition, however, since we could merely set
[edit] Quasiperiodic signals
Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions; instead they in the nature of almost periodic functions and that article should be consulted.
