Difference between revisions of "Quasiperiodicity"
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| − | + | <font color="blue">Quasi-periodicité</font> (''Fr''). <font color="red">Quasiperiodizität</font> (''Ge''). <font color="black">Quasi-periodicità</font> (''It''). <font color="purple">準周期性</font> (''Ja''). <font color="green">Cuasiperiodicidad</font> (''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 '''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 == | |
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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 [ 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'' | ||
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(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. | ||
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| + | [[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.