Multivariate generalized Bernstein polynomials: Identities for orthogonal polynomials of two variables
The authors define polynomials \(B_\mathbf{k}^n(\mathbf{x}; \omega | q)\), which they call multivariate generalized Bernstein polynomials, depending on the two parameters \(q\) and \(\omega\). These polynomials inherit many special cases and identities can be retrieved from limit forms. These polynomials extend the multivariate classical and discrete Bernstein polynomials [see \textit{G. G. Lorentz}, Bernstein polynomials. 2nd ed. New York, NY: Chelsea Publishing. (1986; Zbl 0989.41504)] as well as the univariate \(q\)-Bernstein polynomials introduced by \textit{G. M. Phillips} [Ann. Numer. Math. 4, No. 1--4, 511--518 (1997; Zbl 0881.41008)]. In analogy to the classical multivariate Bernstein polynomials, basic properties of the new polynomials are given, such as recurrence relations with respect to \(\mathbf{k}\) and \(n\), \(q\)-differentiation rules, the de Casteljau algorithm, the partition of unity property and a symmetry property. Furthermore, connections to bivariate big \(q\)-Jacobi polynomials are made. Finally, relations between bivariate \(q\)-Bernstein and little \(q\)-Jacobi polynomials and, respectively, between bivariate discrete Bernstein and Hahn polynomials are given. An application of the polynomials in CAGD is outlined in the Introduction.
scientific article; zbMATH DE number 5549480
scientific article