multiresolution analysis

Definition

A multiresolution analysis is a sequence ${(V_j)}_{j\in \mathbb{Z}}$ of subspaces of $L_2(ℝ)$ such that

1. (nesting) $\mathrm{\dots }\subset V_{-1}\subset V_0\subset V_1\subset \mathrm{\dots }$

2. (density) $\overline{span{\bigcup }_{j\in \mathbb{Z}}{V}_j}=L_2(ℝ)$

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 ${\{\mathrm{\Phi }(t-m)\}}_{m\in \mathbb{Z}}\}$ 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.