Quasiperiodicity

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[[Quasiperiodicity]]
 
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<Font color="blue">Quasi-periodicit&eacute; </font>(Fr.)
<Font color="blue">Quasi-periodicit&eacute; </font>(Fr.)
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'''Definition'''
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== Definition ==
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A function is called  ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks
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A function is called  ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks on positions
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on positions
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<math> k~=~\sum_{i=1}^n h_i  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>
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for basis vectors  '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors
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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
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form a basis for the space (''n'' equal to the space dimension, and linearly
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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''.
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independent basis vectors over the real numbers) then the function is lattice periodic.
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If ''n'' is larger than the space dimension, then the function is  ''aperiodic''.
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'''Comment'''
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== 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.
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  <math>f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( 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''
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'''a''' is smaller than the chosen quantity:
'''a''' is smaller than the chosen quantity:
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| <math>f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>  
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<math>| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>  
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A quasiperiodic function is always an almost periodic function, but the converse
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A quasiperiodic function is always an almost periodic function, but the converse is not true.  
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is not true.  
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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

 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.