Jacobi-weighted orthogonal polynomials on triangular domains

Abstract

We construct Jacobi-weighted orthogonal polynomials 𝒫n,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 𝒫n,r(α,β,γ)(u,v,w) over the triangular domain T satisfy the following properties: 𝒫n,r(α,β,γ)(u,v,w)∈ℒn,n≥1, r=0,1,…,n, and 𝒫n,r(α,β,γ)(u,v,w)⊥𝒫n,s(α,β,γ)(u,v,w) for r≠s. And hence, 𝒫n,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 of the Bernstein polynomial basis.