Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

AbstractThis paper generalizes the mixed extension principle in L2(ℝd) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces Hs(ℝd) and H−s(ℝd). In terms of masks for φ,ψ1,…,ψL∈Hs(ℝd) and $\tilde{\phi},\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}\in H^{-s}({\mathbb{R}}^{d})$ , simple sufficient conditions are given to ensure that (Xs(φ;ψ1,…,ψL), $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ forms a pair of dual wavelet frames in (Hs(ℝd),H−s(ℝd)), where $$\begin{array}{ll}X^{s}\bigl(\phi;\psi^{1},\ldots,\psi^{L}\bigr):=&\bigl\{\phi(\cdot-k):k\in {\mathbb{Z}}^{d}\bigr\}\\[9pt]&{}\cup\bigl\{2^{j(d/2-s)}\psi^{\ell}(2^{j}\cdot-k):j\in {\mathbb{N}}_{0},\ k\in{\mathbb{Z}}^{d},\ \ell=1,\ \ldots,L\bigr\}.\end{array}$$ For s>0, the key of this general mixed extension principle is the regularity of φ, ψ1,…,ψL, and the vanishing moments of $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ , while allowing $\tilde{\phi}$ , $\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}$ to be tempered distributions not in L2(ℝd) and ψ1,…,ψL to have no vanishing moments. So, the systems Xs(φ;ψ1,…,ψL) and $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L})$ may not be able to be normalized into a frame of L2(ℝd). As an example, we show that {2j(1/2−s)Bm(2j⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in Hs(ℝ) for any 0<s<m−1/2, where Bm is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases $(X^{s}(\phi;\psi^{1},\ldots,\psi^{L}),X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ in Sobolev spaces (Hs(ℝd),H−s(ℝd)). For example, all interpolatory wavelet systems in (Donoho, Interpolating wavelet transform. Preprint, 1997) generated by an interpolatory refinable function φ∈Hs(ℝ) with s>1/2 are Riesz bases of the Sobolev space Hs(ℝ). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L2(ℝd), which is quite different from other approaches in the literature.