On the convergence and iterates of q-Bernstein polynomials

Oruc H., Tuncer N.

JOURNAL OF APPROXIMATION THEORY, vol.117, no.2, pp.301-313, 2002 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 117 Issue: 2
  • Publication Date: 2002
  • Doi Number: 10.1006/jath.2002.3703
  • Journal Name: JOURNAL OF APPROXIMATION THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.301-313
  • Keywords: q-Bernstein polynomials, Stirling polynomials, iterates of the q-Bernstein operator, interpolation, LINEAR-COMBINATIONS, OPERATORS, LIMITS
  • Dokuz Eylül University Affiliated: Yes

Abstract

The convergence properties of q-Bernstein polynomials are investigated. When q greater than or equal to 1 is fixed the generalized Bernstein polynomials B(n)f of f, a one parameter family of Bernstein polynomials, converge to f as n --> infinity if f is a polynomial. It is proved that, if the parameter 0 f if and only if f is linear. The iterates of B(n)f are also considered. It is shown that B(n)(M)f 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 --> infinity. Moreover, the iterates of the Boolean sum of B(n)f converge to the interpolating polynomial for f at n + 1 geometrically spaced nodes on [0,1] (C) 2002 Elsevier Science (USA).