Corrected subdivision approximation of piecewise smooth functions

Subdivision schemes are useful mathematical tools for the generation of curves and surfaces. The linear approaches suffer from Gibbs oscillations when approximating functions with singularities. On the other hand, when we analyze the convergence of nonlinear subdivision schemes the regularity of the limit function is smaller than in the linear case. The goal of this paper is to introduce a corrected implementation of linear interpolatory subdivision schemes addressing both properties at the same time: regularity and adaption to singularities with high order of accuracy. In both cases of point-value data and of cell-average data, we are able to construct a subdivision-based algorithm producing approximations with high precision and limit functions with high piecewise regularity and without diffusion nor oscillations in the presence of discontinuities.