On Pseudo Splines

The first type of pseudo splines were introduced by Daubechies,Han,Ron and Shen to construct tight wavelets frames with high approximation order.They were introduced by Selesnick independently.Consider symmetry,Dong and Shen constructed the second type of pseudo splines.Pseudo splines provides a rich family of refinable function.B-splines,orthogonal refinable functions and interpolation refinable functions are special cases of pseudo splines.With the masks of pseudo splines,Han and Shen constructed C^∞nonstationary refinable functions with compact support and give a complete analysis of nonstationary tight wavelets frames. In this paper,we investigate some properties of pseudo splines as following,·Give a regularity analysis of pseudo splines;·Construct symmetric complex pseudo splines and symmetric or antisymmetric complex tight wavelets frames with high approximation order in L₂(R);·Combine the definitions of B-splines and Box splines,we define pseudo box splines.Pseudo box splines are one multivariate generalization of univariate pseudo splines.·With the masks of pseudo splines,we construct nonstationary symmetric C^∞complex pseudo splines and pseudo box splines.Moreover,we investigate the convergence of cascade algorithm associated with the masks h(o|¨)lder continuous.We construct Riesz wavelet bases in L₂(R) with fast decay.We also investigate the support of a refinable vector satisfying an inhomogeneous refinement equation.By using some methods introduced by So and Wang, an estimate is given for the support of each component function of a compactly supported refinable vector satisfying an inhomogeneous matrix refinement equation with finitely supported masks.