Approximation and Orthogonality on Fully Symmetric Domains

ABSTRACT We study orthogonal polynomials on a fully symmetric planar domain that is generated by a certain triangle in the first quadrant. For a family of weight functions on , we show that orthogonal polynomials that are even in the second variable on can be identified with orthogonal polynomials on the unit disk composed with a quadratic map, and the same phenomenon can be extended to the domain generated by the rotation of in higher dimensions. The connection allows an immediate deduction of results for approximation and Fourier orthogonal expansions on these fully symmetric domains. It applies, for example, to analysis on a sector of a disk in two variables and the conic domain derived from the rotation of a sector in high dimensions.