Riesz multiwavelet bases generated by vector refinement equation

Abstract

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L2(ℝs). Suppose ψ = (ψ1,..., ψr)T and \( \tilde \psi = (\tilde \psi ^1 ,...,\tilde \psi ^r )^T \) 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 {ψjkl: l = 1,...,r, j e ℤ, k e ℤs} and \( \tilde \psi _{jk}^\ell :\ell = 1,...,r,j \in \mathbb{Z},k \in \mathbb{Z}^s \) to form two Riesz sequences for L2(ℝs), where ψjkl = mj/2ψl(Mj·−k) and \( \tilde \psi _{jk}^\ell = m^{{j \mathord{\left/ {\vphantom {j 2}} \right. \kern-\nulldelimiterspace} 2}} \tilde \psi ^\ell (M^j \cdot - k) \), M is an s × s integer matrix such that limn→∞M−n = 0 and m = |detM|. Furthermore, let ϕ = (ϕ1,...,ϕr)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 {ψjkνl: ν = 1,..., m − 1, l = 1,...,r, j e ℤ, k e ℤs} and {\( \tilde \psi _{jk}^\nu \) : ν=1,...,m−1,l=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.