×

Jacobi polynomials in Bernstein form. (English) Zbl 1111.33004

Summary: The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so-called Bernstein-Bézier form of a polynomial.

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, 1972 (10th printing).; M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, 1972 (10th printing).
[2] Appell, P.; Kampè de Fériet, J., Fonctions Hypergéomtriques et Hypersphériques—Polynômes d’Hermite (1926), Gauthier-Villars: Gauthier-Villars Paris · JFM 52.0361.13
[3] Berens, H.; Schmid, H. J.; Xu, Y., Bernstein-Durrmeyer polynomials on a simplex, J. Approx. Theory, 68, 247-261 (1992) · Zbl 0754.41005
[4] de Boor, C., B-form basics, (Farin, G., Geometric Modelling: Algorithms and New Trends (1987), SIAM: SIAM Philadelphia)
[5] Braess, D.; Schwab, C., Approximation on simplices with respect to Sobolev norms, J. Approx. Theory, 103, 329-337 (2000) · Zbl 0948.41004
[6] Dahmen, W.; Micchelli, C. A., Convexity and Bernstein polynomials on \(k\)-simploids, Acta Math. Appl. Sinica, 6, 50-66 (1990) · Zbl 0805.41004
[7] Dahmen, W.; Micchelli, C. A.; Seidel, H. P., Blossoming begets B-spline bases built better by B-patches, Math. Comp., 59, 199, 97-115 (1992) · Zbl 0757.41014
[8] Derriennic, M. M., On multivariate approximation by Bernstein-type polynomials, J. Approx. Theory, 45, 325-343 (1985)
[9] Dunkl, C.; Xu, Y., Orthogonal Polynomials in Several Variables (2001), Cambridge University Press: Cambridge University Press Cambridge
[10] Farouki, R. T.; Goodman, T. N.T.; Sauer, T., Construction of an orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Comp. Aided Geom. Design, 20, 209-230 (2003) · Zbl 1069.65517
[11] Farouki, R. T.; Rajan, V. T., On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design, 4, 191-216 (1987) · Zbl 0636.65012
[12] Sauer, T., The genuine Bernstein-Durrmeyer operator on a simplex, Results Math., 26, 99-130 (1994) · Zbl 0817.41014
[13] Sauer, T., Computational aspects of multivariate polynomial interpolation, Adv. Comput. Math., 3, 3, 219-238 (1995) · Zbl 0831.65006
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.