A novel interpolation-regression approach for function approximation on the disk and its application to cubature formulas

TL;DR

This paper introduces a polynomial interpolation-regression method for function approximation on disks, emphasizing node selection for stability, and applies it to develop accurate cubature formulas for disk integration.

Contribution

It presents a new interpolation-regression approach tailored for disk domains, improving stability and accuracy in function approximation and numerical integration.

Findings

Enhanced stability through node selection for Zernike polynomials

Accurate cubature formulas derived for disk integration

Improved function reconstruction on disk domains

Abstract

The interpolation-regression approximation is a powerful tool in numerical analysis for reconstructing functions defined on square or triangular domains from their evaluations at a regular set of nodes. The importance of this technique lies in its ability to avoid the Runge phenomenon. In this paper, we present a polynomial approximation method based on an interpolation-regression approach for reconstructing functions defined on disk domains from their evaluations at a general set of sampling points. Special attention is devoted to the selection of interpolation nodes to ensure numerical stability, particularly in the context of Zernike polynomials. As an application, the proposed method is used to derive accurate cubature formulas for numerical integration over the disk.