multiresolution analysis
Definition
A multiresolution analysis is a sequence ${({V}_{j})}_{j\in \mathbb{Z}}$ of subspaces^{} of ${L}_{2}(\mathbb{R})$ such that

1.
(nesting) $\mathrm{\dots}\subset {V}_{1}\subset {V}_{0}\subset {V}_{1}\subset \mathrm{\dots}$

2.
(density) $\overline{span{\bigcup}_{j\in \mathbb{Z}}{V}_{j}}={L}_{2}(\mathbb{R})$

3.
(separation^{}) ${\bigcap}_{j\in \mathbb{Z}}{V}_{j}=\{0\}$

4.
(scaling) $f(x)\in {V}_{j}$ if and only if $f({2}^{j}x)\in {V}_{0}$

5.
(orthonormal basis^{}) there exists a function $\mathrm{\Phi}\in {V}_{0}$, called a scaling function, such that the system ${\{\mathrm{\Phi}(tm)\}}_{m\in \mathbb{Z}}\}$ is an orthonormal basis in ${V}_{0}.$
Notes
Multiresolution analysis, particularly scaling functions, are used to derive wavelets. The ${V}_{j}$ are called approximation spaces. Several choices of scaling functions may exist for a given set of approximation spaces— each determines a unique multiresolution analysis.
Title  multiresolution analysis 

Canonical name  MultiresolutionAnalysis 
Date of creation  20130322 14:26:48 
Last modified on  20130322 14:26:48 
Owner  swiftset (1337) 
Last modified by  swiftset (1337) 
Numerical id  5 
Author  swiftset (1337) 
Entry type  Definition 
Classification  msc 46C99 
Synonym  level of detail 
Related topic  Wavelet 
Defines  scaling function 