Fully accurate approximation of piecewise smooth functions using corrected B-spline quasi-interpolants

This article is devoted to introducing new spline quasi-interpolants for the sharp approximation of data in one and several dimensions. The new construction aims to obtain accurate approximations close to singularities of the function from which the data are obtained. The technique relies in an accurate knowledge of the position of the singularity, which can be known or approximated, and that allows for the obtention of accurate approximations of the jumps of the function and its derivatives at this point. With this information, it is possible to compute correction terms for the B-spline bases of the spline that are affected by the presence of singularities. The result is a piecewise smooth reconstruction with full accuracy (meaning by full accuracy, the accuracy of the spline at smooth zones). The numerical experiments presented support the theoretical results obtained.