Biorthogonal multiple wavelets generated by vector refinement equation

• DOI: 10.1007/s11425-007-0056-x
• Publication Date: 2007-05-18T16:00:51Z
• Authors: Song Li; Jun Xian
• Keywords: Multiresolution analysis; Matrix (chemical analysis); Sequence (biology)

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ϕ = (ϕ 1, …, ϕ r)T is in $$(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $$ is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M −n = 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.