Abstract
For nonhomogeneous wavelet bi-frames in a pair of dual spaces (H^{s}(mathbb{R}^{d}), H^{-s}(mathbb{R}^{d})) with sneq 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}(mathbb{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.
Highlights
1 Introduction Most classical nonhomogeneous wavelet systems are derived from a refinable structure
Observe that for wavelet systems derived from refinable structures, one of the most important features is their associated fast wavelet transform
This paper addresses nonhomogeneous wavelet bi-frames under the setting of reducing subspaces of Hs(Rd) which is more general than Hs(Rd)
Summary
Most classical nonhomogeneous wavelet systems are derived from a refinable structure (see [2, 5, 7, 9, 21] and the references therein). In this paper, we will dicuss the nonhomogeneous wavelet bi-frames under the setting of the reducing subspaces of Sobolev spaces. Li and Zhang in [18] obtained the following characterization for a nonhomogeneous wavelet bi-frames of (Hs(Rd), H–s(Rd)). Proposition 1.1 Let Xs(ψ0; ) and X–s(ψ 0; ̃ ) be Bessel sequences in Hs(Rd) and H–s(Rd), respectively.
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