The Approximation of All Continuous Functions on [0, 1] by q-Bernstein Polynomials in the Case q → 1+

Since for q > 1, the q-Bernstein polynomials Bn,q(f;.) are not positive linear operators on C[0,1], their convergence properties are not similar to those in the case 0 < q ≤ 1. It has been known that, in general, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{n,q_{n}} (f; .)$$\end{document} does not approximate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C[0, 1]$$\end{document} if qn → 1+, n → ∞, unlike in the case qn → 1−. In this paper, it is shown that if 0 ≤ qn − 1 = o(n−1 3−n), n → ∞, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C[0, 1]$$\end{document}, we have: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{n,q_{n}} (f; x) \rightarrow f(x)$$\end{document} as n → ∞, uniformly on [0,1].