# Quasiperiodicity

### From Online Dictionary of Crystallography

Quasi-periodicité (*Fr*). Quasiperiodizität (*Ge*). Quasi-periodicità (*It*). 準周期性 (*Ja*). Cuasiperiodicidad (*Sp*).

## Definition

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

[math] k~=~\sum_{i=1}^n h_i 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(r)~=~\sum_k A(k) \cos [ 2\pi k. r+\varphi (k) ]. [/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(r+ a)-f( r) |~\lt ~ \varepsilon~~{\rm for ~all~ 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.