Polynomial mapped bases: theory and applications

TL;DR

This paper introduces polynomial mapped bases, a novel technique for scattered data interpolation and related applications, which effectively mitigates Runge's and Gibbs effects by incorporating data discontinuities through a specialized mapping function.

Contribution

It develops the basic theory of polynomial mapped bases and demonstrates their effectiveness in various applications like interpolation, quadrature, and bio-imaging reconstruction.

Findings

Reduces Runge's and Gibbs effects significantly

Enables incorporation of data discontinuities

Improves accuracy in scattered data applications

Abstract

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects.