Jacobi-weighted orthogonal polynomials on triangular domains

Abstract

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