  The mapped bases or Fake Nodes Approach (FNA), introduced in [10], allows to
change the set of nodes without the need of resampling the function. Such
scheme has been successfully applied in preventing the appearance of the Gibbs
phenomenon when interpolating discontinuous functions. However, the originally
proposed S-Gibbs map suffers of a subtle instability when the interpolant is
constructed at equidistant nodes, due to the Runge's phenomenon. Here, we
propose a novel approach, termed Gibbs-Runge-Avoiding Stable Polynomial
Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated.
After providing a theoretical analysis of the Lebesgue constant associated to
the mapped nodes, we test the new approach by performing different numerical
experiments which confirm the theoretical findings.
