Explicit formula relating the Jacobi, Hahn and Bernstein polynomials
SIAM Journal on Mathematical Analysis
Discrete Be´zier curves and surfaces
Mathematical methods in computer aided geometric design II
Discrete Bernstein bases and Hahn polynomials
SPOA VII Proceedings of the seventh Spanish symposium on Orthogonal polynomials and applications
Journal of Approximation Theory
Convergent inversion approximations for polynomials in Bernstein form
Computer Aided Geometric Design
Legendre-Bernstein basis transformations
Journal of Computational and Applied Mathematics - Special issue/Dedicated to Prof. Larry L. Schumaker on the occasion of his 60th birthday
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code
Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code
Application of Legendre--Bernstein basis transformations to degree elevation and degree reduction
Computer Aided Geometric Design
q-Bernstein polynomials and Bézier curves
Journal of Computational and Applied Mathematics
Connections between two-variable Bernstein and Jacobi polynomials on the triangle
Journal of Computational and Applied Mathematics
Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials
Computer Aided Geometric Design
Two-variable orthogonal polynomials of big q-Jacobi type
Journal of Computational and Applied Mathematics
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The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63-78] Bin(x; ω|q):= 1/(ωq)n[n, i]q xi(ωx-1;q)i(x; q)n-i (i=0,1,...,n). We give explicitly the dual basis functions Dkn(x; a, b, ω| q) for the polynomials Bin(x; ω| q), in terms of big q-Jacobi polynomials Pk (x; a, b, ω/q; q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula--relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials--is also given. Further, an alternative formula is given, representing the dual polynomial Djn (0 ≤ j ≤ n) as a linear combination of min(j, n - j) + 1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by Dkn, as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311-346].