Iterates of (?, q)-Bernstein operators
FILOMAT, vol.37, no.22, pp.7663-7671, 2023 (SCI-Expanded, Scopus)
- 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.