Actions

Difference between revisions of "Quasiperiodicity"

From Online Dictionary of Crystallography

(Tidied translations and added German and Spanish (U. Mueller))
 
(4 intermediate revisions by 2 users not shown)
Line 1: Line 1:
[[Quasiperiodicity]]
+
<font color="blue">Quasi-periodicit&eacute;</font> (''Fr''). <font color="red">Quasiperiodizität</font> (''Ge''). <font color="black">Quasi-periodicit&agrave;</font> (''It''). <font color="purple">準周期性</font> (''Ja''). <font color="green">Cuasiperiodicidad</font> (''Sp'').
  
 +
== Definition ==
  
<Font color="blue">Quasi-periodicit&eacute; </font>(Fr.)
+
A function is called  ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks on positions
  
  '''Definition'''
+
<math> k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)  </math>
  
A function is called ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks
+
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
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''.
  
<math> k~=~\sum_{i=1}^n h_i  a_i^*,~~({\rm integers ~}h_i)  </math>
+
== Comment ==
 
 
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
 
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:
  
  <math>f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( k) \right). </math>
+
<math>f(r)~=~\sum_k A(k) \cos [ 2\pi  k. r+\varphi (k) ]. </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''
Line 29: Line 23:
 
'''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>  
+
<math>| f(r+ a)-f( r) |~<~ \varepsilon~~{\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.
 +
 +
[[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

[math] k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i) [/math]

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:

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

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:

[math]| f(r+ a)-f( r) |~\lt ~ \varepsilon~~{\rm for ~all~ r} .[/math]

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.