Difference between revisions of "Quasiperiodicity"
From Online Dictionary of Crystallography
TedJanssen (talk | contribs) |
BrianMcMahon (talk | contribs) m |
||
| Line 1: | Line 1: | ||
| − | |||
| − | |||
| − | |||
<Font color="blue">Quasi-periodicité </font>(Fr.) | <Font color="blue">Quasi-periodicité </font>(Fr.) | ||
| − | + | == Definition == | |
| − | A function is called ''quasiperiodic'' if its Fourier transform consists of δ-peaks | + | A function is called ''quasiperiodic'' if its Fourier transform consists of δ-peaks on positions |
| − | on positions | ||
| − | + | <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 form a basis for the space (''n'' equal to the space dimension, and linearly |
| − | 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''. |
| − | 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. | ||
| Line 22: | Line 16: | ||
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> | |
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> | |
| − | 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. | ||
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
[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 \left( 2\pi k. r+\phi ( k) \right). [/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 ~ \epsilon~~{\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.