A characterization of nonhomogeneous wavelet bi-frames for reducing subspaces of Sobolev spaces

For nonhomogeneous wavelet bi-frames in a pair of dual spaces (H-s(R-d), H-s(R-d)) with s not equal 0, smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other one for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in L-2(R-d). In this paper, we introduce the reducing subspaces of Sobolev spaces, and characterize the nonhomogeneous wavelet bi-frames under the setting of a general pair of dual reducing subspaces of Sobolev spaces.