Riesz multiwavelet bases generated by vector refinement equation  

作　　者：LI Song LIU ZhiSong 

机构地区：[1]Department of Mathematics,Zhejiang University,Hangzhou 310027,China [2]School of Mathematics,Physics&Information Science,Zhejiang Ocean University,Zhoushan 316004,China

出　　处：《Science China Mathematics》2009年第3期468-480,共13页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)

摘　　要：In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L 2(? s ). Suppose ψ = (ψ1,..., ψ r ) T and $ \tilde \psi = (\tilde \psi ^1 ,...,\tilde \psi ^r )^T $ are two compactly supported vectors of functions in the Sobolev space (H μ(? s )) r for some μ > 0. We provide a characterization for the sequences {ψ jk l : l = 1,...,r, j ε ?, k ε ? s } and $ \tilde \psi _{jk}^\ell :\ell = 1,...,r,j \in \mathbb{Z},k \in \mathbb{Z}^s $ to form two Riesz sequences for L 2(? s ), where ψ jk l = m j/2ψ l (M j ·?k) and $ \tilde \psi _{jk}^\ell = m^{{j \mathord{\left/ {\vphantom {j 2}} \right. \kern-0em} 2}} \tilde \psi ^\ell (M^j \cdot - k) $ , M is an s × s integer matrix such that lim n→∞ M ?n = 0 and m = |detM|. Furthermore, let ? = (?1,...,? r ) T and $ \tilde \phi = (\tilde \phi ^1 ,...,\tilde \phi ^r )^T $ be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, $ \tilde a $ and M, where a and $ \tilde a $ are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr ) T and $ \tilde \psi ^\nu = (\tilde \psi ^{\nu 1} ,...,\tilde \psi ^{\nu r} )^T $ , ν = 1,..., m ? 1 such that two sequences {ψ jk νl : ν = 1,..., m ? 1, l = 1,...,r, j ε ?, k ε ? s } and { $ \tilde \psi _{jk}^\nu $ : ν=1,...,m?1,?=1,...,r, j ∈ ?, k ∈ ? s } form two Riesz multiwavelet bases for L 2(? s ). The bracket product [f, g] of two vectors of functions f, g in (L 2(? s )) r is an indispensable tool for our characterization.In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(Rs). Suppose ψ = (ψ1, . . . , ψr)T and ψ = ( ψ1, . . . , ψr)T are two compactly supported vectors of functions in the Sobolev space (Hμ(Rs))r for some μ > 0. We provide a characterization for the sequences {ψjk : = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jk : = 1, . . . , r, j ∈ Z, k ∈ Zs} to form two Riesz sequences for L2(Rs), where ψjk = mj/2ψ (M j ·k) and ψjk = mj/2 ψ (M j ·k), M is an s × s integer matrix such that limn→∞ Mn = 0 and m = |detM|. Furthermore, let = (1, . . . , r)T and = ( 1, . . . , r)T be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, a and M, where a and a are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1, . . . , ψνr)T and ψν = ( ψν1, . . . , ψ?νr)T , ν = 1, . . . , m 1 such that two sequences {ψjνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} and {ψ jνk : ν = 1, . . . , m 1, = 1, . . . , r, j ∈ Z, k ∈ Zs} form two Riesz multiwavelet bases for L2(Rs). The bracket product [f, g] of two vectors of functions f, g in (L2(Rs))r is an indispensable tool for our characterization.

关 键 词：vector refinement equations Riesz multiwavelet base biorthogonal wavelets 42C40 39B12 46B15 47A10 47B37 

分 类 号：O174.41[理学—数学]