# Quasiperiodicity

### From Online Dictionary of Crystallography

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<Font color="blue">Quasi-periodicité </font>(Fr.) | <Font color="blue">Quasi-periodicité </font>(Fr.) | ||

- | + | == Definition == | |

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

- | 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 | + | 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 |

- | 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''. |

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

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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 \left( 2\pi k. r+\phi ( k) \right). </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) |~<~ \epsilon~~{\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. |

## Revision as of 17:28, 7 February 2012

Quasi-periodicité (Fr.)

## 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.