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Vector subdivision schemes in \((L_p(\mathbb R^s))^r\) (\(1\leq p\leq\infty\)) spaces. (English) Zbl 1218.42018

Summary: The purpose of this paper is to investigate refinement equations of the form \[ \varphi(x)=\sum_{a \in \mathbb Z^{s}} a(\alpha)\varphi(Mx - a),~ x\in \mathbb R^{s}, \] where the vector of functions \(\varphi =(\varphi _{1}\dots , \varphi _{ r })^{ T }\) is in \((L_{ p }(\mathbb R^{ s }))^{ r }\), \(1\leq p\leq\infty\), \(a(\alpha )\), \(\alpha \in \mathbb Z^{ s }\), is a finitely supported sequence of \(r\times r\) matrices called the refinement mask, and \(M\) is an \(s \times s\) integer matrix such that \(\lim_{n\rightarrow \infty} M^{-n}= 0\). In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions \(\varphi _{0}\in (L _{ p }(\mathbb R^{ s }))^{ r }\) and use the iteration schemes \(f_{ n }:=Q_{a}^{n} \varphi _{0}\), \(n=1,2,\dots \), where \(Q_{ n }\) is the linear operator defined on \((L _{ p }(\mathbb R^{ s }))^{ r }\) by \[ Q_{a}\varphi: = \sum_{a \in \mathbb Z^{s}} a(\alpha)\varphi(M \cdot - a),~\varphi \in (L_{p} (\mathbb R^{s}))^{r}. \] This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the \(L_{p}\)-convergence of subdivision schemes in terms of the \(p\)-norm joint spectral radius of a finite collection of some linear operators determined by the sequence \(a\) and the set \(B\) restricted to a certain invariant subspace, where the set \(B\) is a complete set of representatives of the distinct cosets of the quotient group \(\mathbb Z^{s}/M\mathbb Z^{s}\) containing \(0\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
47N40 Applications of operator theory in numerical analysis
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