Biorthogonal multiple wavelets generated by vector refinement equation

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 where the vector of functions ϕ = (ϕ ₁, …, ϕ _r)^T is in 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. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10071071 and 10471123), the Mathematical Tianyuan Foundation of the National Natural Science Foundation of China NSF (Grant No. 10526036) and China Postdoctoral Science Foundation (Grant No. 20060391063)