In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quasiperiodic function is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that has a certain similarity to a periodic function. A function
is quasiperiodic with quasiperiod
if
, where
is a "''simpler''" function than
. What it means to be "''simpler''" is vague.
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:
:
Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:
:
An example of this is the
Jacobi theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
, where
:
shows that for fixed
it has quasiperiod
; it also is periodic with period one. Another example is provided by the
Weierstrass sigma function
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and \wp functions is analogou ...
, 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
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and \wp functions is analogou ...
, where
:
for a ''z''-independent η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where
we say ''f'' is
periodic with period ω in the period lattice
.
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
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Haral ...
s and that article should be consulted. The more vague and general notion of
quasiperiodicity
Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpred ...
has even less to do with quasiperiodic functions in the mathematical sense.
A useful example is the function:
:
If the ratio ''A''/''B'' is rational, this will have a true period, but if ''A''/''B'' is irrational there is no true period, but a succession of increasingly accurate "almost" periods.
See also
*
Quasiperiodic motion
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.
That is, if we imagine that the phase space ...
External links
Quasiperiodic functionat
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Complex analysis
Types of functions