Evaluating Polynomials Over the Unit Disk and the Unit Ball

TL;DR

This paper explores efficient methods for evaluating orthonormal polynomials over the unit disk and ball, and applies these to polynomial approximation of functions, providing practical algorithms and MATLAB code.

Contribution

It introduces an efficient evaluation technique for orthonormal polynomial bases over the unit disk and ball, facilitating polynomial approximation tasks.

Findings

Efficient evaluation algorithms for orthonormal polynomials over B_2 and B_3.

Method for least squares approximation of functions using these polynomials.

Provision of MATLAB codes for practical implementation.

Abstract

We investigate the use of orthonormal polynomials over the unit disk B_2 in R^2 and the unit ball B_3 in R^3. An efficient evaluation of an orthonormal polynomial basis is given, and it is used in evaluating general polynomials over B_2 and B_3. The least squares approximation of a function f on the unit disk by polynomials of a given degree is investigated, including how to write a polynomial using the orthonormal basis. Matlab codes are given.