Iterates of (?, q)-Bernstein operators

Koroglu B., TAŞDELEN YEŞİLDAL F.

FILOMAT, vol.37, no.22, pp.7663-7671, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 37 Issue: 22
  • Publication Date: 2023
  • Doi Number: 10.2298/fil2322663k
  • Journal Name: FILOMAT
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
  • Page Numbers: pp.7663-7671
  • Keywords: q-calculus, Iterates, (?, q)-Bernstein polynomials, Uniform convergence
  • Ankara University Affiliated: Yes

Abstract

In this paper, the iterates of (alpha, q)-Bernstein operators are considered. Given fixed n E N and q > 0, it is shown that for f E C[0, 1] the k-th iterate Tkn,q,alpha(f; x) converges uniformly on [0,1] to the linear function Lf(x) passing through the points (0, f (0)) and (1, f (1)). Moreover, it is proved that, when q E (0, 1), the iterates Tjnn,q,alpha (f; x), in which {jn}-* oo as n-* oo, also converge to Lf(x). Further, when q E (1, oo) and {jn} is a sequence of positive integers such that jn/[n]q-* t as n-* oo, where 0 <= t <= oo, the convergence of the iterates Tjnn,q,alpha(p; x) for p being a polynomial is studied.