A class of multiwavelets and projected frames from two-direction wavelets

This article aims at studying two-direction refinable functions and two-direction wavelets in the setting ?^s, s > 1. We give a sufficient condition for a two-direction refinable function belonging to L²(?^s). Then, two theorems are given for constructing biorthogonal (orthogonal) two-direction refinable functions in L²(?^s) and their biorthogonal (orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal (orthogonal) two-direction wavelets, symmetric biorthogonal (orthogonal) multiwaveles in L²(?^s can be obtained easily. Applying the projection method to biorthogonal (orthogonal) two-direction wavelets in L²(?^s, we can get dual (tight) two-direction wavelet frames in L²(?^m, where. m ≥ s From the projected dual (tight) two-direction wavelet frames in L²(?^m, symmetric dual (tight) frames in L²(?^m can be obtained easily. In the end, an example is given to illustrate theoretical results.