Difference between revisions of "Quasiperiodicity"
From Online Dictionary of Crystallography
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| − | <Font color="blue">Quasi-periodicité </font>(''Fr'') | + | <Font color="blue">Quasi-periodicité </font>(''Fr''). <Font color="black">Quasi-periodicità </font>(''It''). <Font color="purple">準周期性 </font>(''Ja''). |
== Definition == | == Definition == | ||
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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 | + | <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'' | ||
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'''a''' is smaller than the chosen quantity: | '''a''' is smaller than the chosen quantity: | ||
| − | <math>| f(r+ a)-f( r) |~<~ \ | + | <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
[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.