Quasiperiodicity

Quasi-periodicité (Fr);　Quasi-periodicità (It);　準周期性 (Ja).

Definition

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

$k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$

for basis vectors ai* 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.

$f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$

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:

$| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$

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.