Quasiperiodicity

Quasiperiodicity

Quasi-periodicité (Fr.)

Definition

A function is called quasiperiodic if its Fourier transform consists of δ-peaks on positions

[math]{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)  [/math]

for basis vectors a_i^* in a space of dimension m. If the basis vectors form a basis for the space (n equal to the space dimension, and linearly independent basis vectors over the real numbers) then the function is lattice periodic. If n is larger than the space dimension, then the function is aperiodic.

Comment

Sometimes the definition includes that the function is not lattice periodic.

A quasiperiodic function may be expressed in a convergent trigonometric series.

 [math]f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right). [/math]

It is a special case of an almost periodic function. An almost periodic function is a function f(r) such that for every small number ε there is a translation a such that the difference between the function and the function shifted over

a is smaller than the chosen quantity:

| [math]f({\bf r}+{\bf a})-f({\bf r}) |~\lt ~ \epsilon~~{\rm for ~all~{\bf r}} .[/math] 

A quasiperiodic function is always an almost periodic function, but the converse is not true.

The theory of almost-periodic functions goes back to the work by H. Bohr.