Multivariate approximation at fake nodes

The main goal of the present paper is to extend the interpolation via the so-called mapped bases without resampling to any basis and dimension. So far indeed, we investigated the univariate case, its extension to rational polynomial interpolation and its natural application to numerical integration. The concept of mapped bases has been widely studied, but all the proposed methods show convergence provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. Thus, we propose an effective method for interpolating via mapped bases in the multivariate setting. We might refer to the method as Fake Nodes Approach (FNA). Our theoretical results are confirmed by various numerical experiments devoted to point out the robustness of the proposed scheme.