# Quasiperiodicity

### From Online Dictionary of Crystallography

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- | + | <font color="blue">Quasi-periodicité</font> (''Fr''). <font color="red">Quasiperiodizität</font> (''Ge''). <font color="black">Quasi-periodicità</font> (''It''). <font color="purple">準周期性</font> (''Ja''). <font color="green">Cuasiperiodicidad</font> (''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'''<sub>i</sub><sup>*</sup> 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 == | |

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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 [ 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'' | ||

is a function ''f''('''r''') such that for every small number ε there is | 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 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) |~<~ \varepsilon~~{\rm for ~all~ r} .</math> | |

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

- | is not true. | + | |

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

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+ | [[Category: Fundamental crystallography]] |

## Latest revision as of 10:16, 17 November 2017

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

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.