On the convergence and iterates of q-Bernstein polynomials

Oruc H., Tuncer N.

JOURNAL OF APPROXIMATION THEORY, cilt.117, sa.2, ss.301-313, 2002 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 117 Sayı: 2
  • Basım Tarihi: 2002
  • Doi Numarası: 10.1006/jath.2002.3703
  • Dergi Adı: JOURNAL OF APPROXIMATION THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.301-313
  • Anahtar Kelimeler: q-Bernstein polynomials, Stirling polynomials, iterates of the q-Bernstein operator, interpolation
  • Dokuz Eylül Üniversitesi Adresli: Evet

Özet

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).