# Quasiperiodicity

### From Online Dictionary of Crystallography

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- | <Font color="blue">Quasi-periodicité </font>(''Fr'') | + | <Font color="blue">Quasi-periodicité </font>(''Fr''). <Font color="black">Quasi-periodicità </font>(''It''). <Font color="purple">準周期性 </font>(''Ja''). |

== Definition == | == Definition == | ||

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

- | A quasiperiodic function may be expressed in a convergent trigonometric series | + | A quasiperiodic function may be expressed in a convergent trigonometric series: |

- | <math>f( r)~=~\sum_k A(k) \cos | + | <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'' | It is a special case of an almost periodic function. An ''almost periodic function'' | ||

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'''a''' is smaller than the chosen quantity: | '''a''' is smaller than the chosen quantity: | ||

- | <math>| f(r+ a)-f( r) |~<~ \ | + | <math>| f(r+ a)-f( r) |~<~ \varepsilon~~{\rm for ~all~ r} .</math> |

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

## Revision as of 13:49, 16 May 2017

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

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:

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:

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.