Stable discontinuous mapped bases: the Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA) method

TL;DR

The paper introduces GRASPA, a new polynomial approximation method that mitigates both Gibbs and Runge phenomena, providing stable interpolation for discontinuous functions without resampling.

Contribution

The paper proposes the GRASPA method, combining mapped bases to prevent Gibbs and Runge phenomena, with theoretical analysis and numerical validation.

Findings

The Lebesgue constant analysis confirms stability of the mapped nodes.

Numerical experiments demonstrate improved approximation of discontinuous functions.

GRASPA effectively prevents Gibbs and Runge phenomena in polynomial interpolation.

Abstract

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.