Evaluating Polynomials Over the Unit Disk and the Unit Ball

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.