We construct Jacobi-weighted orthogonal polynomials P_(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 p_(n,r)~(α,β,γ) (u,v,w) over the triangular domain T satisfy the following properties: p_(n,r)~(α,β,γ) (u,v,w),r = 0,1,...,n,andp_(n,r)~(α,β,γ) (u,v,w)(w,v,w) p_(n,r)~(α,β,γ) (u,v,w)(u,v,w)forr 4s.Hence,<3>{n/'yu,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.
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