# Quasiperiodicity

(Difference between revisions)
 Revision as of 18:28, 18 May 2009 (view source)← Older edit Revision as of 17:28, 7 February 2012 (view source)mNewer edit → Line 1: Line 1: - [[Quasiperiodicity]] - - Quasi-periodicité (Fr.) Quasi-periodicité (Fr.) - '''Definition''' + == 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 + - $k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$ + $k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$ - for basis vectors  '''a'''i* in a space of dimension ''m''. If the basis vectors + for basis vectors  '''a'''i* 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''' + == 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. - $f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$ + $f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$ 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: - | $f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$ + $| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$ - 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

$k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$

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.

$f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$

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:

$| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$

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.