# Quasiperiodicity

### From Online Dictionary of Crystallography

**Quasiperiodicity**

Quasi-periodicité (Fr.)

Definition

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

Failed to parse (syntax error): {\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)

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.

Failed to parse (syntax error): f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf 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

ais smaller than the chosen quantity:

|Failed to parse (syntax error): f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf 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.