# Quasiperiodicity

(Difference between revisions)
 Revision as of 15:49, 18 May 2009 (view source)← Older edit Latest revision as of 10:16, 17 November 2017 (view source) (Tidied translations and added German and Spanish (U. Mueller)) (5 intermediate revisions not shown) Line 1: Line 1: - [[Quasiperiodicity]] + Quasi-periodicité (''Fr''). Quasiperiodizität (''Ge''). Quasi-periodicità (''It''). 準周期性 (''Ja''). Cuasiperiodicidad (''Sp''). + == Definition == - Quasi-periodicité (Fr.) + A function is called  ''quasiperiodic'' if its Fourier transform consists of δ-peaks on positions - '''Definition''' + $k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$ - A function is called ''quasiperiodic'' if its Fourier transform consists of δ-peaks + 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 - on positions + 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''. - ${\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)$ + == Comment == - + - 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. 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: - $f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right).$ + $f(r)~=~\sum_k A(k) \cos [ 2\pi k. r+\varphi (k) ].$ 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: + '''a''' is smaller than the chosen quantity: - | $f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf r}} .$ + $| f(r+ a)-f( r) |~<~ \varepsilon~~{\rm for ~all~ r} .$ - 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. + + [[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

$k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$

for basis vectors ai* 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:

$f(r)~=~\sum_k A(k) \cos [ 2\pi k. r+\varphi (k) ].$

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:

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

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.