Dual generalized Bernstein basis

  • Authors:

  • Stanisław Lewanowicz;Paweł Woźny

  • Affiliations:

  • Institute of Computer Science, University of Wrocław, ul. Przesmyckiego 20, 51-151 Wrocław, Poland;Institute of Computer Science, University of Wrocław, ul. Przesmyckiego 20, 51-151 Wrocław, Poland

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2006

Quantified Score

Hi-index 0.00

Visualization

Abstract

The generalized Bernstein basis in the space @P"n 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. Wozny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63-78], B"i^n(x;@w|q)@?1(@w;q)"nni"qx^i(@wx^-^1;q)"i(x;q)"n"-"i(i=0,1,...,n).We give explicitly the dual basis functions D"k^n(x;a,b,@w|q) for the polynomials B"i^n(x;@w|q), in terms of big q-Jacobi polynomials P"k(x;a,b,@w/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 D"j^n(0=