Abstract
In order to get divergence free and curl free wavelets, one introduced the smoothed pseudo spline by using the convolution method. The smoothed pseudo splines can be considered as an extension of pseudo splines. In this paper, we first show that the shifts of a smoothed pseudo spline are linearly independent. The linear independence of the shifts of a pseudo spline is a necessary and sufficient condition for the construction of the biorthogonal wavelet system. Based on this result, we generalize the results of Riesz wavelets and derive biorthogonal wavelets from smoothed pseudo splines. Furthermore, by applying the unitary extension principle, we construct tight frame systems associated with smoothed pseudo splines with desired approximation order.
Highlights
In order to construct tight framelets with desired approximation orders, the first type of pseudo splines was first introduced in [ ] and [ ]
By using the unitary extension principle, we get the construction of tight framelets with desired approximation order based on smoothed pseudo splines
4 The construction of biorthogonal wavelet based on the results of Section, we focus on the construction of biorthogonal wavelets from the smoothed pseudo splines
Summary
In order to construct tight framelets with desired approximation orders, the first type of pseudo splines was first introduced in [ ] and [ ]. By using the unitary extension principle, we get the construction of tight framelets with desired approximation order based on smoothed pseudo splines. The refinement mask of the first type of pseudo splines with order (m, l) is given by L m + l sin j ξ cos (l–j) ξ , j and the refinement mask of the second type of pseudo splines with order (m, l) is given by
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