Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L 2(ℝ d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H s (ℝ d ) and H −s (ℝ d ). In terms of masks for φ,ψ 1,…,ψ L ∈H s (ℝ d ) and $\tilde{\phi},\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}\in H^{-s}({\mathbb{R}}^{d})$ , simple sufficient conditions are given to ensure that (X s (φ;ψ 1,…,ψ L ), $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L}))$ forms a pair of dual wavelet frames in (H s (ℝ d ),H −s (ℝ d )), 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, ψ 1,…,ψ L , 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 L 2(ℝ d ) and ψ 1,…,ψ L to have no vanishing moments. So, the systems X s (φ;ψ 1,…,ψ L ) and $X^{-s}(\tilde{\phi};\tilde{\psi}^{1},\ldots,\tilde{\psi}^{L})$ may not be able to be normalized into a frame of L 2(ℝ d ). As an example, we show that {2 j(1/2−s) B m (2 j ⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in H s (ℝ) for any 01/2 are Riesz bases of the Sobolev space H s (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 L 2(ℝ d ), which is quite different from other approaches in the literature.