Editing Interval (mathematics) (section)

==Dyadic intervals==

A ''dyadic interval'' is a bounded real interval whose endpoints are <math>\tfrac{j}{2^n}</math> and <math>\tfrac{j+1}{2^n},</math> where <math>j</math> and <math>n</math> are integers. Depending on the context, either endpoint may or may not be included in the interval.

Dyadic intervals have the following properties:

* The length of a dyadic interval is always an integer power of two.
* Each dyadic interval is contained in exactly one dyadic interval of twice the length.
* Each dyadic interval is spanned by two dyadic intervals of half the length.
* If two open dyadic intervals overlap, then one of them is a subset of the other.

The dyadic intervals consequently have a structure that reflects that of an infinite [[binary tree]].

Dyadic intervals are relevant to several areas of numerical analysis, including [[adaptive mesh refinement]], [[multigrid methods]] and [[wavelet|wavelet analysis]]. Another way to represent such a structure is [[p-adic analysis]] (for {{math|1=''p'' = 2}}).<ref>{{cite journal |last1=Kozyrev |first1=Sergey |year=2002 |title=Wavelet theory as {{mvar|p}}-adic spectral analysis |journal=[[Izvestiya: Mathematics|Izvestiya RAN. Ser. Mat.]] |volume=66 |issue=2 |pages=149–158 |doi=10.1070/IM2002v066n02ABEH000381 |url=http://mi.mathnet.ru/eng/izv/v66/i2/p149 |access-date=2012-04-05|arxiv=math-ph/0012019 |bibcode=2002IzMat..66..367K |s2cid=16796699 }}</ref>