# Quasiperiodicity

(Difference between revisions)
 Revision as of 07:57, 26 February 2015 (view source)m (cat)← Older edit Revision as of 13:49, 16 May 2017 (view source)m (Style edits to align with printed edition)Newer edit → Line 1: Line 1: - Quasi-periodicité (''Fr'');　Quasi-periodicità (''It'');　準周期性 (''Ja''). + Quasi-periodicité (''Fr''). Quasi-periodicità (''It''). 準周期性 (''Ja''). == Definition == == Definition == Line 14: Line 14: 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: - $f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right).$ + $f(r)~=~\sum_k A(k) \cos [ 2\pi k. r+\varphi (k) ].$ 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 23: Line 23: '''a''' is smaller than the chosen quantity: '''a''' is smaller than the chosen quantity: - $| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .$ + $| f(r+ a)-f( r) |~<~ \varepsilon~~{\rm for ~all~ r} .$ 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 $k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)$

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: $f(r)~=~\sum_k A(k) \cos [ 2\pi k. r+\varphi (k) ].$

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

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.