Functions with discontinuities appear in many application such as image reconstruction, interface problems, and etc. Accurate approximation and interpolation of these functions are therefore of great importance.
After giving an introduction on the required notions, we present an approximation method of discontinuous function f, which incorporates the discontinuities into the approximant s. In a nutshell, the idea is to control the influence of the data on the approximant, not only with regards to their distance, but also with regards to the discontinuities of the underlying function. The numerical experiments show an improvement on the accuracy of the approximation compared with the conventional schemes.