# 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. 1.

(nesting) $\ldots\subset V_{-1}\subset V_{0}\subset V_{1}\subset\ldots$

2. 2.

(density) $\overline{\mathop{\rm span}\bigcup_{j\in\mathbb{Z}}V_{j}}=L_{2}({\mathbb{R}})$

3. 3.

(separation) $\bigcap_{j\in\mathbb{Z}}V_{j}=\{0\}$

4. 4.

(scaling) $f(x)\in V_{j}$ if and only if $f(2^{-j}x)\in V_{0}$

5. 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 MultiresolutionAnalysis 2013-03-22 14:26:48 2013-03-22 14:26:48 swiftset (1337) swiftset (1337) 5 swiftset (1337) Definition msc 46C99 level of detail Wavelet scaling function