Vector cascade algorithms with infinitely supported masks in weighted L2-spaces
Vector cascade algorithms with infinitely supported masks in weighted L2-spaces
复制标题
DOI：
10.1007/s10114-012-1295-5
复制
发表时间：
2012-12
期刊：
影响因子：
--
通讯作者：
Jian Bin Yang
中科院分区：
文献类型：
--
作者：
Jian Bin Yang
In this paper, we shall study the solutions of functional equations of the formwhere Φ = (ϕ1, ...,ϕr)Tis anr× 1 column vector of functions on thes-dimensional Euclidean space,is an exponentially decaying sequence ofr×rcomplex matrices called refinement mask andMis ans × sinteger matrix such that limn→ ∞M−n= 0. We are interested in the question, for a maskawith exponential decay, if there exists a solution Φ to the functional equation with each functionϕj,j= 1, ...,r, belonging toL2(ℝs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weightedL2spaces. The vector cascade operatorQa,Massociated with maskaand matrixMis defined by $$Q_{a,M} f: = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )f(M \cdot - \alpha ), f = \left( {f_1 , \ldots f_r } \right)^T \in \left( {L_{2,\mu } \left( {\mathbb{R}^s } \right)} \right)^r .}$$ The iterative scheme (Qa,Mnf)n=1,2,...is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2(ℝs))r, the weightedL2space. Inspired by some ideas in [Jia, R. Q., Li, S.: Refinable functions with exponential decay: An approach via cascade algorithms.J. Fourier Anal. Appl.,17, 1008–1034 (2011)], we prove that if the vector cascade algorithm associated withaandMconverges in (L2(ℝs))r, then its limit function belongs to (L2,μ(ℝs))rfor some µ > 0.
In this paper, we shall study the solutions of functional equations of the formwhere Φ = (ϕ1, ...,ϕr)Tis anr× 1 column vector of functions on thes-dimensional Euclidean space,is an exponentially decaying sequence ofr×rcomplex matrices called refinement mask andMis ans × sinteger matrix such that limn→ ∞M−n= 0. We are interested in the question, for a maskawith exponential decay, if there exists a solution Φ to the functional equation with each functionϕj,j= 1, ...,r, belonging toL2(ℝs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weightedL2spaces. The vector cascade operatorQa,Massociated with maskaand matrixMis defined by $$Q_{a,M} f: = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )f(M \cdot - \alpha ), f = \left( {f_1 , \ldots f_r } \right)^T \in \left( {L_{2,\mu } \left( {\mathbb{R}^s } \right)} \right)^r .}$$ The iterative scheme (Qa,Mnf)n=1,2,...is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2(ℝs))r, the weightedL2space. Inspired by some ideas in [Jia, R. Q., Li, S.: Refinable functions with exponential decay: An approach via cascade algorithms.J. Fourier Anal. Appl.,17, 1008–1034 (2011)], we prove that if the vector cascade algorithm associated withaandMconverges in (L2(ℝs))r, then its limit function belongs to (L2,μ(ℝs))rfor some µ > 0.