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

DOI10.1016/S0167-8396(03)00025-6zbMath1069.65517OpenAlexW1996988461MaRDI QIDQ2388505

Rida T. Farouki, Thomas Sauer, Tim N. T. Goodman

Publication date: 14 September 2005

Published in: Computer Aided Geometric Design (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s0167-8396(03)00025-6

zbMATH Keywords

orthogonal polynomialsLegendre polynomialsbarycentric coordinatestriangular domainsBernstein representationhigher-dimensional simplexes

Mathematics Subject Classification ID

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis (42C05) Computer-aided design (modeling of curves and surfaces) (65D17)

Related Items (27)

Linearizations of matrix polynomials in Bernstein bases ⋮ Jacobi polynomials in Bernstein form ⋮ Functional data approximation on bounded domains using polygonal finite elements ⋮ Unisolvency for polynomial interpolation in simplices with symmetrical nodal distributions ⋮ A novel high-performance quadrature rule for BEM formulations ⋮ Differentiation matrices in polynomial bases ⋮ Bounds-constrained polynomial approximation using the Bernstein basis ⋮ Boundary element dynamical energy analysis: a versatile method for solving two or three dimensional wave problems in the high frequency limit ⋮ High-precision computation of the weak Galerkin methods for the fourth-order problem ⋮ A Framework for Structured Linearizations of Matrix Polynomials in Various Bases ⋮ Approximate implicitization using linear algebra ⋮ Connections between two-variable Bernstein and Jacobi polynomials on the triangle ⋮ Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis ⋮ Bivariate orthogonal polynomials on triangular domains ⋮ Bernstein collocation method for neutral type functional differential equation ⋮ On the formulation of a BEM in the Bézier-Bernstein space for the solution of Helmholtz equation ⋮ Fast inversion of the simplicial Bernstein mass matrix ⋮ Recurrence relations for orthogonal polynomials on triangular domains ⋮ Gauss-Lobatto to Bernstein polynomials transformation ⋮ A vertex-based hierarchical slope limiter for \(p\)-adaptive discontinuous Galerkin methods ⋮ Constrained degree reduction of polynomials in Bernstein-Bézier form over simplex domain ⋮ GPU-Accelerated Bernstein--Bézier Discontinuous Galerkin Methods for Wave Problems ⋮ On the Bernstein--Bézier form of Jacobi polynomials on a simplex ⋮ Three term recurrence for the evaluation of multivariate orthogonal polynomials ⋮ Bivariate generalized shifted Gegenbauer orthogonal system ⋮ An accurate treatment of non-homogeneous boundary conditions for development of the BEM ⋮ Evaluation of the derivative of a polynomial in Bernstein form

Cites Work

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• Chebyshev economization for parametric surfaces
• On multivariate approximation by Bernstein-type polynomials
• On the numerical condition of polynomials in Bernstein form
• Algorithms for polynomials in Bernstein form
• The genuine Bernstein-Durrmmeyer operator on a simplex
• Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle
• Legendre-Bernstein basis transformations
• Convergent inversion approximations for polynomials in Bernstein form
• Orthogonal polynomials in two variables
• Orthogonal Polynomials on the Hexagon
• The Recursion Formulas for Orthogonal Polynomials innVariables
• Orthogonality and Recursion Formulas for Polynomials in n Variables
• Recurrence Formulas for Multivariate Orthogonal Polynomials
• Unbounded Commuting Operators and Multivariate Orthogonal Polynomials
• Multivariate Orthogonal Polynomials and Operator Theory
• On the optimal stability of the Bernstein basis
• Note on Orthogonal Polynomials inv-Variables
• On Multivariate Orthogonal Polynomials

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