Journal of Applied Mathematics
Volume 2005 (2005), Issue 3, Pages 205-217
doi:10.1155/JAM.2005.205

Jacobi-weighted orthogonal polynomials on triangular domains

A. Rababah and M. Alqudah

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

Received 25 March 2004; Revised 20 March 2005

Copyright © 2005 A. Rababah and M. Alqudah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.