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 $ ...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.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)
文摘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.
