In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation ...In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial An （g） converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.展开更多
基金Supported by National Natural Science Foundation of China （Grant No. 61063020）, Ningxia Ziran （Grant No. NZ0907） and 2009 Ningxia Gaoxiao Foundations We would like to thank two anonymous referees for their insightful comments.
文摘In this paper, two ways of the proof are given for the fact that the Bernstein-Bezier coefficients （BB-coefficients） of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial An （g） converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.
