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) $\ldots\subset V_{-1}\subset V_{0}\subset V_{1}\subset\ldots$
2.

(density) $\overline{\mathop{\rm span}\bigcup_{j\in\mathbb{Z}}V_{j}}=L_{2}({\mathbb{R}})$
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 $\Phi\in V_{0}$ , called a scaling function, such that the system $\{\Phi(t-m)\}_{m\in\mathbb{Z}}\}$ 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	2013-03-22 14:26:48
Last modified on	2013-03-22 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