Jacobi-weighted orthogonal polynomials on triangular domains

Abstract

We construct Jacobi-weighted orthogonal polynomials ℘n,r(α,β,γ) (u, v, w), α, β, γ > - 1, α + β + γ = 0, on the triangular domain T. We show that these polynomials ℘n,r(α,β,γ) (u, v, w) over the triangular domain T satisfy the following properties: ℘n,r(α,β,γ) (u, v, w) ∈ ℒn, n ≥ 1, r = 0, 1,..., n, and ℘n,r(α,β,γ) (u, v, w) ⊥ ℘n,r(α,β,γ) (u, v, w) for r ≠ s. And hence, ℘n,r(α,β,γ) (u, v, w), n = 0, 1, 2,..., r = 0, 1,..., n form, an orthogonal system over the triangular domain T with respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties ofthe Bernstein polynomial basis.