multiresolution analysisMultiresolutionAnalysis2004-06-24 20:00:402004-06-25 10:17:16Definitionscaling function% this is the default PlanetMath preamble. as your knowledge
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\paragraph{Definition}
A \emph{multiresolution analysis} is a sequence $(V_j)_{j\in \mathbb Z}$ of subspaces of $L_2({\mathbb R})$ such that
\begin{enumerate}
\item (nesting) $ \ldots \subset V_{-1} \subset V_0 \subset V_1 \subset \ldots $
\item (density) $\overline{\mathop{\rm span} \bigcup_{j \in \mathbb Z} V_j } = L_2({\mathbb R}) $
\item (separation) $ \bigcap_{j \in \mathbb Z} V_j = \{0\}$
\item (scaling) $f(x) \in V_j$ if and only if $f(2^{-j} x) \in V_0$
\item (orthonormal basis) there exists a function $\Phi \in V_0$, called a \emph{scaling function}, such that the system $\{ \Phi(t -m) \}_{m \in \mathbb Z} \}$ is an orthonormal basis in $V_0.$
\end{enumerate}
\paragraph{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.