Construction of nonseparable dual [formula omitted]-wavelet frames in [formula omitted

Abstract

Suppose { ψ ℓ } ℓ = 1 a 1 , ψ ˜ ℓ ℓ = 1 a 1 and ψ ℓ ♮ ℓ = 1 2 a 2 + 1 , ψ ˜ ℓ ♮ ℓ = 1 2 a 2 + 1 are two pairs of dual M-wavelet frames and N-wavelet frames in L 2 ( R s 1 ) and L 2 ( R s 2 ) , respectively, where M and N are s 1 × s 1 and s 2 × s 2 dilation matrices with a 1 ⩾ ( | det ( M ) | - 1 ) and ( 2 a 2 + 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 L 2 ( R s ) are constructed, where s = s 1 + s 2 , and Ω = M Θ O N with Θ and M - 1 Θ 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 L 2 ( R s ) . From the dual Ω -wavelet frames, nonseparable dual Ω -wavelet frames with symmetry can be obtained easily. In the end, two examples are given.