Arbitrarily smooth orthogonal nonseparable wavelets in R2
SIAM Journal on Mathematical Analysis
Approximation properties and construction of Hermite interpolants and biorthogonal mutliwavelets
Journal of Approximation Theory
Quincunx fundamental refinable functions and Quincunx biorthogonal wavelets
Mathematics of Computation
Vector cascade algorithms and refinable function vectors in Sobolev spaces
Journal of Approximation Theory
Explicit construction of symmetric orthogonal wavelet frames in L2(Rs)
Journal of Approximation Theory
Journal of Approximation Theory
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The dual 2I d -framelets in $ (H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d})) $ , s驴驴0, were introduced by Han and Shen (Constr Approx 29(3):369---406, 2009). In this paper, we systematically study the Bessel property of multiwavelet sequences in Sobolev spaces. The conditions for Bessel multiwavelet sequence in $ H^{-s}(\mathbb{R}^{d}) $ take great difference from those for Bessel wavelet sequence in this space. Precisely, the Bessel property of multiwavelet sequence in $ H^{-s}(\mathbb{R}^{d}) $ is not only related to multiwavelets themselves but also to the corresponding refinable function vector. We construct a class of Bessel M-refinable function vectors with M being an isotropic dilation matrix, which have high Sobolev smoothness, and of which the mask symbols have high sum rules. Based on the constructed Bessel refinable function vector, an explicit algorithm is given for dual M-multiframelets in $ (H^{s}(\mathbb{R}^{d}),H^{-s}(\mathbb{R}^{d})) $ with the multiframelets in $ H^{-s}(\mathbb{R}^{d}) $ having high vanishing moments. On the other hand, based on the dual multiframelets, an algorithm for dual M-multiframelets with symmetry is given. In Section 6, we give an example to illustrate the constructing procedures of dual multiframelets.