On Sobolev orthogonal polynomials on a triangle

Abstract

We use the invariance of the triangle T 2 = { ( x , y ) ∈ R 2 : 0 ⩽ x , y , 1 − x − y } \mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\} under the permutations of { x , y , 1 − x − y } \{x,y,1-x-y\} to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on T 2 \mathbf {T}^2 . These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.