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)n,n1, r=0,1,,n, and 𝒫n,r(α,β,γ)(u,v,w)𝒫n,s(α,β,γ)(u,v,w) for rs. 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.