We construct Jacobi-weighted orthogonal polynomials
$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w), \ \alpha,\beta,\gamma > -1, \ \alpha+\beta+\gamma=0$ , on the triangular domain $T$ . We show that these polynomials $\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)$ over the triangular domain $T$ satisfy the following properties: $\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)\in {\mathcal L}_{n}$, $n\geq 1$ , $r=0,1,\dotsc,n,$ and ${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w) \perp {\mathcal P}_{n,s}^{(\alpha,\beta,\gamma)}(u,v,w)$ for $r\neq s$ . And hence, ${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w)$ , $n=0,1,2,\ldots$ , $r=0,1,\dotsc,n$ 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.