On Sobolev orthogonal polynomials on a triangle

Abstract:

We use the invariance of the triangle $\mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\}$ under the permutations of $\{x,y,1-x-y\}$ to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on $\mathbf {T}^2$. 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.