Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder

TL;DR

This paper investigates polynomial approximation and numerical integration on a cylindrical domain using Approximate Fekete and Leja points derived from Weakly Admissible Meshes, demonstrating their effectiveness through analysis of Lebesgue constants and errors.

Contribution

It introduces a method for selecting approximation and cubature points on a cylinder using AFP and DLP from WAMs, showing their suitability for accurate polynomial approximation and integration.

Findings

AFP and DLP from WAMs are effective for polynomial approximation.

Lebesgue constants grow at a controlled rate, indicating stable approximation.

Approximation and cubature errors are minimized using these points.

Abstract

The paper deals with polynomial interpolation, least-square approximation and cubature of functions defined on the rectangular cylinder, $K=D\times [-1,1]$, with $D$ the unit disk. The nodes used for these processes are the {\it Approximate Fekete Points} (AFP) and the {\it Discrete Leja Points} (DLP) extracted from suitable {\it Weakly Admissible Meshes} (WAMs) of the cylinder. From the analysis of the growth of the Lebesgue constants, approximation and cubature errors, we show that the AFP and the DLP extracted from WAM are good points for polynomial approximation and numerical integration of functions defined on the cylinder.