Quasiperiodicity

From Online Dictionary of Crystallography

(Difference between revisions)
Jump to: navigation, search
m (cat)
m (Style edits to align with printed edition)
Line 1: Line 1:
-
<Font color="blue">Quasi-periodicit&eacute; </font>(''Fr''); <Font color="black">Quasi-periodicit&agrave; </font>(''It''); <Font color="purple">準周期性 </font>(''Ja''). 
+
<Font color="blue">Quasi-periodicit&eacute; </font>(''Fr''). <Font color="black">Quasi-periodicit&agrave; </font>(''It''). <Font color="purple">準周期性 </font>(''Ja''). 
== Definition ==
== Definition ==
Line 14: Line 14:
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 23: 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 is not true.  
A quasiperiodic function is always an almost periodic function, but the converse is not true.  

Revision as of 13:49, 16 May 2017

Quasi-periodicité (Fr). Quasi-periodicità (It). 準周期性 (Ja). 

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.