When approximation order is an odd positive integer, a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex wavelets. In the end, there are several examples that illustrate the corresponding results.
Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a(r+s) ×(r+s) dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr+s/Zr+s areΓΩ={[γT h,ζ T q] T:γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton's condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2(Rr+s).Properties,including Lawton's condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2(Rr+1) are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].
This article aims at studying two-direction refinable functions and two-direction wavelets in the setting Rs, s > 1. We give a sufficient condition for a two-direction refinable function belonging to L2(Rs). Then, two theorems are given for constructing biorthogonal(orthogonal) two-direction refinable functions in L2(Rs) and their biorthogonal(orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal(orthogonal)two-direction wavelets, symmetric biorthogonal(orthogonal) multiwaveles in L2(Rs) can be obtained easily. Applying the projection method to biorthogonal(orthogonal) two-direction wavelets in L2(Rs), we can get dual(tight) two-direction wavelet frames in L2(Rm), where m ≤ s. From the projected dual(tight) two-direction wavelet frames in L2(Rm), symmetric dual(tight) frames in L2(Rm) can be obtained easily. In the end, an example is given to illustrate theoretical results.
