Fourier analysis of 2-point hermite interpolatory subdivision schemes

Abstract

Two subdivision schemes with Hermite data on Z are studied. These schemes use 2 or 7 parameters respectively depending on whether Hermite data involve only rst derivatives or include second derivatives. For a large region in the parameter space, the schemes are convergent in the space of Schwartz distributions. The Fourier transform of any i n terpolating function can be computed through products of matrices of order 2 or 3. The Fourier transform is related to a speci c system of functional equations whose analytic solution is unique except for a multiplicative constant. The main arguments for these results come from Paley-Wiener-Schwartz theorem on the characterization of the Fourier transforms of distributions with compact support and a theorem of Artzrouni about convergent products of matrices.Math Subject Classi cation: 40A20, 42A38, 46F12, 65D05, 65D10

A circle-preserving Hermite interpolatory subdivision scheme with tension control

“…For any other positive value of w, a detailed smoothness analysis should be performed. But, because the refinement equations of this Hermite scheme are not in the same form of the ones presented and analyzed in the literature up to now (Dyn and Levin, 1999;Dubuc et al, 2001;Yu, 2005;Dubuc, 2006), at this moment there are not the required tools to give a precise answer. However, all our experimental results let us conjecture that curvature continuity is preserved for any choice of w > 0.…”

“…This section is devoted to the illustration of some numerical examples confirming the effectiveness of the proposed 2-point Hermite-interpolatory subdivision scheme (Figs. 4,5,6,7,8,9). Due to the fact that the novel refinement rules are built upon a piecewise C 2 Hermite interpolant, they should naturally guarantee a C 2 limit curve for any positive value of w. For the particular starting polylines in Fig.…”

Two order C² Hermite interpolatory subdivision

“…Juttler analyzed convergence and continuity of Hermite subdivision of any order [3]. Dubuc discussed first and second order Hermite subdivision schemes based on Fourier methods [4]. L.Romani discussed a circle-preserving second order Hermite interpolatory subdivision scheme [5].…”