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multiresolution analysis
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(Definition)
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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 $V_0.$
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.
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"multiresolution analysis" is owned by swiftset.
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See Also: wavelet
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level of detail |
| Also defines: |
scaling function |
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Cross-references: approximation, wavelets, orthonormal basis, scaling, subspaces, sequence
There are 2 references to this entry.
This is version 2 of multiresolution analysis, born on 2004-06-24, modified 2004-06-25.
Object id is 5961, canonical name is MultiresolutionAnalysis.
Accessed 4967 times total.
Classification:
| AMS MSC: | 46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous) |
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Pending Errata and Addenda
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