Dual generalized Bernstein basis

نویسندگان

  • Stanislaw Lewanowicz
  • Pawel Wozny
چکیده

The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced recently by G.M. Phillips, is given by the formula (see S. Lewanowicz & P. Woźny, BIT 44 (2004), 63–78) Bn i (x;ω| q) := 1 (ω; q)n [ n i ] q x (ωx−1; q)i (x; q)n−i (i = 0, 1, . . . , n). We give explicitly the dual basis functions Dn k (x; a, b, ω| q) for the polynomials Bn i (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 Dn j (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 Dn k , as well as an identity which may be seen as an analogue of the extended Marsden’s identity.

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عنوان ژورنال:
  • Journal of Approximation Theory

دوره 138  شماره 

صفحات  -

تاریخ انتشار 2006