Semi-regular Dubuc-Deslauriers wavelet tight frames

In this paper, we construct wavelet tight frames with n vanishing moments for Dubuc-Deslauriers 2npoint semi-regular interpolatory subdivision schemes. Our motivation for this construction is its practical use for further regularity analysis of wide classes of semi-regular subdivision. Our constructive tools are local eigenvalue convergence analysis for semi-regular Dubuc-Deslauriers subdivision, the Unitary Extension Principle and the generalization of the Oblique Extension Principle to the irregular setting by Chui, He and St\"ockler. This group of authors derives suitable approximation of the inverse Gramian for irregular Bspline subdivision. Our main contribution is the derivation of the appropriate approximation of the inverse Gramian for the semi-regular Dubuc-Deslauriers scaling functions ensuring n vanishing moments of the corresponding framelets

A piecewise polynomial approach to analyzing interpolatory subdivision

The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative