Dual generalized Bernstein basis |
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Authors: | Stanis aw Lewanowicz,Pawe Wo
ny |
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Affiliation: | Institute of Computer Science, University of Wrocław, ul. Przesmyckiego 20, 51-151 Wrocław, Poland |
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Abstract: | 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], We give explicitly the dual basis functions for the polynomials , 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 (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 , 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]. |
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Keywords: | Generalized Bernstein basis q-Bernstein basis Bernstein basis Discrete Bernstein basis Dual basis Big q-Jacobi polynomials Little q-Jacobi polynomials Shifted Jacobi polynomials |
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