Quasiperiodicity

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on positions
on positions
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  <math>{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)  </math>
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  <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
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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.
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   <math>f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right). </math>
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   <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''
is a function ''f''('''r''') such that for every small number &epsilon; there is
is a function ''f''('''r''') such that for every small number &epsilon; 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
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'''a''' is smaller than the chosen quantity:
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'''a''' is smaller than the chosen quantity:
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  | <math>f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf r}} .</math>  
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  | <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

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.