Abstract
In this paper, two ways of the proof are given for the fact that the Bernstein-Bézier 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 A n (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.
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Bini, D. A., Gemignani, L., Winkler, J. R.: Structured matrix methods for CAGD: An application to computing the resultant of polynomials in the Bernstein basis. Numer. Linear Algebra Appl., 12, 685–698 (2005)
Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design, A.K. Peters, Wellesley, 1993
Lorentz, G.: Bernstein Polynomials, Toronto Press, Toronto, 1953
Othman, W. A. M., Goldman, R. N.: The dual basis functions for the generalized Ball basis of odd degree. Comput. Aided Geom. Design, 14, 571–582 (1997)
Zhao, K., Sun, J.: Dual bases of multivariate Bernstein-Bézier polynomials. Comput. Aided Geom. Design, 5, 119–125 (1988)
Li, F. J., Xu, Z. B., Zheng, K. J.: Optimal approximation order for q-Stancu operators defined on a simplex. Acta Mathematica Sinica, Chinese Series, 51, 135–144 (2008)
Li, F. J., Xu, Z. B.: The approximation properties of generalized multivatiate Bernstein operators. Acta Mathematicae Applicatae Sinica, Chinese Series, 30, 955–960 (2008)
Barry, P. J., Goldman, R. N.: What Is the Natural Generalization of a Bézier Curve? In: Mathematical Methods in Computer Aided Geometric Design, Academic Press, Boston, 1989
Trump, W., Prautzsch, H.: Arbitrarily high degree elevation of Bézier representations. Comput. Aided Geom. Design, 13, 387–398 (1996)
Farouki, R., Rajan, V. T.: Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Design, 5, 1–26 (1988)
Cohen, E., Schumaker, L. L.: Rate of convergence of controlpolygons. Comput. Aided Geom. Design, 2, 305–322 (1985)
Farin, G.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Design, 3, 83–127 (1986)
Prautzsch, H.: On convex Bézier triangles. Math. Mod. Numer. Anal., 26, 23–26 (1992)
Neamtu, M.: Subdividing multivariate polynomials over simplices in Bernstein-Bézier form without de Casteljau algorithm. In: Curves and Surface, Academic Press, Boston, 1991
Chui, C. K., Li, X.: Realization of neural networks with one hidden layer. In: Multivariate Approximation: From CAGD to Wavelets, World Scientific, Singapore, 1993
Sablonnière, P.: Discrete Bézier Curves and Surface. In: Mathematical Methods in Computer Aided Geometric Design II, Academic Press, Boston, 1992
Floater, M. S., Lyche, T.: Asymptotic convergence of degree-raising. Adv. Comput. Math., 12, 175–187 (2000)
Moret, I., Novati, P.: Interpolating functions of matrices on of quasi-kernel polynomials. Numer. Linear Algebra Appl., 12, 337–353 (2005)
De Boor, C., Fix, G.: Spline interpolation by quasi-interpolants. J. Approx. Theory, 8, 19–45 (1973)
Hakopian, H. A.: On a class of Hermite interpolation problems. Adv. Comput. Math., 12, 303–309 (2000)
Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math., 2, 377–410 (2000)
De Boor, C.: On interpolation by radial polynomials. Adv. Comput. Math., 24, 143–153 (2006)
De Boor, C.: B-form Basis. In: Geometric Modeling: Algorithms and New Trends, SIAM, Philadelphia, 1987, 131–148
Gander, W.: Change of basis in polynomial interpolation. Numer. Linear Algebra Appl., 12, 769–778 (2005)
Chung, K. C., Yao, T. H.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal., 14, 735–741 (1977)
Lee, S. L., Phillips, G. M.: Construction of lattices for Lagrange interpolation in projective space. Constr. Approx., 7, 283–297 (1991)
Sauer, T., Xu, Y.: On multivariate Lagrange interpolation. Math. Comp., 64, 1147–1170 (1995)
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Supported by National Natural Science Foundation of China (Grant No. 61063020), Ningxia Ziran (Grant No. NZ0907) and 2009 Ningxia Gaoxiao Foundations
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Li, F.J. Interpolation and convergence of Bernstein-Bézier coefficients. Acta. Math. Sin.-English Ser. 27, 1769–1782 (2011). https://doi.org/10.1007/s10114-011-8462-y
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DOI: https://doi.org/10.1007/s10114-011-8462-y