Difference between revisions of "Quasiperiodicity"
From Online Dictionary of Crystallography
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on positions | on positions | ||
| − | <math> | + | <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. | ||
| − | <math>f( | + | <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 ε there is | 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 translation '''a''' such that the difference between the function and the function shifted over | ||
| − | + | '''a''' is smaller than the chosen quantity: | |
| − | | <math>f( | + | | <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
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.