PlanetMath: multiresolution analysis

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multiresolution analysis

(Definition)

Definition

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

(nesting) $\ldots \subset V_{-1} \subset V_0 \subset V_1 \subset \ldots$
(density) $\overline{\mathop{\rm span} \bigcup_{j \in \mathbb{Z}} V_j } = L_2({\mathbb{R}})$
(separation) $\bigcap_{j \in \mathbb{Z}} V_j = \{0\}$
(scaling) $f(x) \in V_j$ if and only if $f(2^{-j} x) \in V_0$
(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

Notes

Multiresolution analysis, particularly scaling functions, are used to derive wavelets. The

are called approximation spaces. Several choices of scaling functions may exist for a given set of approximation spaces-- each determines a unique multiresolution analysis.

"multiresolution analysis" is owned by swiftset.

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