# Quasiperiodicity

(Difference between revisions)
 Revision as of 15:49, 18 May 2009 (view source)← Older edit Revision as of 18:28, 18 May 2009 (view source)Newer edit → Line 9: Line 9: on positions on positions - ${\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)$ + $k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$ for basis vectors  '''a'''i* in a space of dimension ''m''. If the basis vectors for basis vectors  '''a'''i* in a space of dimension ''m''. If the basis vectors Line 22: Line 22: A quasiperiodic function may be expressed in a convergent trigonometric series. A quasiperiodic function may be expressed in a convergent trigonometric series. - $f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right).$ + $f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$ 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: - | $f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf r}} .$ + | $f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$ A quasiperiodic function is always an almost periodic function, but the converse A quasiperiodic function is always an almost periodic function, but the converse

## Revision as of 18:28, 18 May 2009

Quasiperiodicity

Quasi-periodicité (Fr.)

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.

 $f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$


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:

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


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.