Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Recurrence Relations for Orthogonal Polynomials on Triangular Domains

Abstract: In Farouki et al, 2003, Legendre-weighted orthogonal polynomials Pn,r(u, v, w), r = 0, 1, . . . , n, n ≥ 0 on the triangular domain T = {(u, v, w) : u, v, w ≥ 0, u+ v+w = 1} are constructed, where u, v, w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is...

Jacobi-weighted Orthogonal Polynomials on Triangular Domains

We construct Jacobi-weighted orthogonal polynomials (α,β,γ) n,r (u,v,w), α,β,γ > −1, α+ β + γ = 0, on the triangular domain T . We show that these polynomials (α,β,γ) n,r (u, v,w) over the triangular domain T satisfy the following properties: (α,β,γ) n,r (u,v,w) ∈ n, n≥ 1, r = 0,1, . . . ,n, and (α,β,γ) n,r (u,v,w) ⊥ (α,β,γ) n,s (u,v,w) for r =s. Hence, (α,β,γ) n,r (u,v,w), n= 0,1,2, . . ., r =...

Asymptotics of Orthogonal Polynomials via Recurrence Relations

Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The o...

Recurrence Formulas for Multivariate Orthogonal Polynomials

In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have dim n¡( common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.