Construction of nonseparable dual Ω-wavelet frames in L2(Rs)

Suppose {ψℓ}ℓ=1a1,ψ˜ℓℓ=1a1 and ψℓ♮ℓ=12a2+1,ψ˜ℓ♮ℓ=12a2+1 are two pairs of dual M-wavelet frames and N-wavelet frames in L2(Rs1) and L2(Rs2), respectively, where M and N are s1×s1 and s2×s2 dilation matrices with a1⩾(|det(M)|-1) and (2a2+1)⩾(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in L2(Rs) are constructed, where s=s1+s2, and Ω=MΘON with Θ and M-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in L2(Rs). From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.