Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains. (English) Zbl 1069.65517

Summary: 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 orthogonal polynomials reduce to the usual Legendre polynomials along one edge of the domain triangle, and within each fixed degree are characterized by vanishing Bernstein coefficients on successive rows parallel to that edge. Closed-form expressions and recursive algorithms for computing the Bernstein coefficients of these orthogonal bivariate polynomials are derived, and their application to surface smoothing problems is sketched. Finally, an extension of the scheme to the construction of orthogonal bases for polynomials over higher-dimensional simplexes is also presented.


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


[1] Appell, P.; Kampé de Fériet, J., Fonctions hypergéometriques at hypersphériques—polynômes d’Hermite, (1926), Gauthier-Villars Paris · JFM 52.0361.13
[2] Askey, R., Orthogonal polynomials and special functions, (1975), SIAM Philadelphia · Zbl 0298.26010
[3] Bertran, M., Note on orthogonal polynomials in ν-variables, SIAM J. math. anal., 6, 250-257, (1975) · Zbl 0273.33010
[4] Davis, P.J., Interpolation and approximation, (1975), Dover New York · Zbl 0111.06003
[5] Derriennic, M.-M., On multivariate approximation by Bernstein-type polynomials, J. approx. theory, 45, 155-166, (1985) · Zbl 0578.41010
[6] Dunkl, C.F., Orthogonal polynomials on the hexagon, SIAM J. appl. math., 47, 343-351, (1987) · Zbl 0613.33010
[7] Dunkl, C.F.; Xu, Y., Orthogonal polynomials of several variables, (2001), Cambridge University Press · Zbl 0964.33001
[8] Farin, G., Triangular bernstein-Bézier patches, Computer aided geometric design, 3, 83-127, (1986)
[9] Farin, G., Curves and surfaces for CAGD, (1993), Academic Press Boston
[10] Farouki, R.T., Convergent inversion approximations for polynomials in Bernstein form, Computer aided geometric design, 17, 179-196, (2000) · Zbl 0939.68126
[11] Farouki, R.T., Legendre – bernstein basis transformations, J. comput. appl. math., 119, 145-160, (2000) · Zbl 0962.65042
[12] Farouki, R.T.; Goodman, T.N.T., On the optimal stability of the Bernstein basis, Math. comp., 65, 1553-1566, (1996) · Zbl 0853.65051
[13] Farouki, R.T.; Rajan, V.T., On the numerical condition of polynomials in Bernstein form, Computer aided geometric design, 4, 191-216, (1987) · Zbl 0636.65012
[14] Farouki, R.T.; Rajan, V.T., Algorithms for polynomials in Bernstein form, Computer aided geometric design, 5, 1-26, (1988) · Zbl 0648.65007
[15] Gould, H.W., Combinatorial identities, (1972), Morgantown W. Va · Zbl 0263.05013
[16] Hoffman, K.; Kunze, R., Linear algebra, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0212.36601
[17] Jackson, D., Formal properties of orthogonal polynomials in two variables, Duke math. J., 2, 423-434, (1936) · JFM 62.0302.02
[18] Kolb, A.; Pottmann, H.; Seidel, H.P., Fair surface reconstruction using quadratic functionals, (), 469-479
[19] Koornwinder, T.H., Two-variable analogues of the classical orthogonal polynomials, () · Zbl 0297.33021
[20] Koornwinder, T.H., 1976. Jacobi polynomials and their two-variable analogues. Thesis, University of Amsterdam
[21] Koornwinder, T.H.; Schwartz, A.L., Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle, Constr. approx., 13, 537-567, (1997) · Zbl 0937.33009
[22] Kowalski, M.A., The recursion formulas for orthogonal polynomials in n variables, SIAM J. math. anal., 13, 309-315, (1982) · Zbl 0494.33011
[23] Kowalski, M.A., Orthogonality and recursion formulas for polynomials in n variables, SIAM J. math. anal., 13, 316-323, (1982) · Zbl 0497.33011
[24] Krall, H.L.; Sheffer, I.M., Orthogonal polynomials in two variables, Ann. mat. pura appl., 76, 325-376, (1967) · Zbl 0186.38602
[25] Lachance, M.A., Chebyshev economization for parametric surfaces, Computer aided geometric design, 5, 195-208, (1988) · Zbl 0709.65012
[26] Proriol, J., Sur une famille de polynomes á deux variables orthogonaux dans un triangle, C. R. acad. sci. Paris, 245, 2459-2461, (1957) · Zbl 0080.05204
[27] Sauer, T., The genuine bernstein – durrmeyer operator on a simplex, Results in mathematics, 26, 99-130, (1994) · Zbl 0817.41014
[28] Szegö, G., Orthogonal polynomials, (1975), American Mathematical Society Providence, RI · JFM 65.0278.03
[29] Xu, Y., On multivariate orthogonal polynomials, SIAM J. math. anal., 24, 783-794, (1993) · Zbl 0770.42016
[30] Xu, Y., Unbounded commuting operators and multivariate orthogonal polynomials, Proc. amer. math. soc., 119, 1223-1231, (1993) · Zbl 0796.33011
[31] Xu, Y., Common zeros of polynomials in several variables and higher dimensional quadrature, (1994), Longman Scientific and Technical Harlow, Essex, England
[32] Xu, Y., Multivariate orthogonal polynomials and operator theory, Trans. amer. math. soc., 343, 193-202, (1994) · Zbl 0832.42017
[33] Xu, Y., Recurrence formulas for multivariate orthogonal polynomials, Math. comp., 62, 687-702, (1994) · Zbl 0802.42021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.