On the convergence and iterates of q-Bernstein polynomials

  • Authors:

  • Halil Oruç;Necibe Tuncer

  • Affiliations:

  • Department of Mathematics, Faculty of Arts and Sciences, Dokuz Eylül University, Tinaztepe, Kampüsü Buca Izmir, Turkey;Department of Mathematics, Faculty of Arts and Sciences, Dokuz Eylül University, Tinaztepe, Kampüsü Buca Izmir, Turkey

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2002

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Abstract

The convergence properties of q-Bernstein polynomials are investigated. When q≥1 is fixed the generalized Bernstein polynomials Bnf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0qBnf→f if and only if f is linear. The iterates of Bnf are also considered. It is shown that BnMf converges to the linear interpolating polynomial for f at the endpoints of [0, 1], for any fixed q 0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of Bnf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0, 1].