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Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

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Abstract

This paper generalizes the mixed extension principle in L 2(ℝd) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H s(ℝd) and H s(ℝd). In terms of masks for φ,ψ 1,…,ψ LH s(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 (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(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, 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(ℝd) 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(ℝd). As an example, we show that {2j(1/2−s) B m (2j⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in H s(ℝ) for any 0<s<m−1/2, where B m 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 (H s(ℝd),H s(ℝd)). For example, all interpolatory wavelet systems in (Donoho, Interpolating wavelet transform. Preprint, 1997) generated by an interpolatory refinable function φH s(ℝ) with s>1/2 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(ℝd), which is quite different from other approaches in the literature.

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Correspondence to Bin Han.

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Communicated by Tim N.T. Goodman.

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Han, B., Shen, Z. Dual Wavelet Frames and Riesz Bases in Sobolev Spaces. Constr Approx 29, 369–406 (2009). https://doi.org/10.1007/s00365-008-9027-x

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