Generalized Bernstein-Chlodowsky polynomials
For given positive integers \(n\) and \(m\), the generalization of Bernstein-Chlodowsky polynomials is defined by \[ B_{n,m}(f,x)= \Biggl( 1+(m-1) \frac{x}{b_n} \Biggr) \sum_{k=0}^{[n/m]} f\Biggl( \frac{b_nk} {n-(m-1)k}\Biggr) C_{n-(m-1)k}^k \Biggl( \frac{x}{b_n} \Biggr)^k \Biggl(1- \frac{x}{b_n} \Biggr)^{n-mk}, \] where \(b_n\) is a sequence of positive numbers such that \(\lim_{n\to\infty} b_n= \infty\), \(\lim_{n\to\infty} (b_n/n)=0\), \(0\leq x\leq b_n\) and \([p]\) denotes the greatest integer less than \(p\). Using a Korovkin type theorem [see \textit{A. D. Gadzhiev}, Math. Zametki 20, 781-786 (1976; Zbl 0383.41016)] the authors obtained a theorem about convergence of \(B_{n,m}(f,x)\) to \(f(x)\) as \(n\to\infty\) in the weighted space of functions \(f\) continuous on the positive semiaxis and satisfying the condition \(\lim_{x\to\infty} (f(x)/ (1+x^2))= k_f<\infty\). But the theorem of convergence rate is not established.
scientific article; zbMATH DE number 1309878