On the rate of convergence of SSB operator and SSK operator for functions of bounded variation (Q1338175)

Let \(f\) be a normalized function of bounded variation on \([0,1]\). Let \[ L_ n(f, x)= \sum^ n_{k=0} f\left({k\over n+ \alpha_ n}\right) q_{n,k,s}(x), \] \[ L^*_ n(f, x)= (\phi_ n+ 1) \sum^ n_{k=0} q_{n,k,s}(x) \int^{(k+1)/(\phi_ n+1)}_{k/(\phi_ n+ 1)} f(t)dt, \] where \[ \begin{aligned} q_{n,k,s}(x) & =\begin{cases} (1-x) P_{n-s,k}(x) & (0\leq k< s),\\ (1- x)P_{n-s,k}(x)+ xP_{n-s,k-s}(x) & (s\leq k\leq n- s),\\ xP_{n-s,k-s}(x) & (n- s< k\leq n)\end{cases}\\ P_{n,k}(x) & =\left(\begin{smallmatrix} n\\ k\end{smallmatrix}\right) x^ k(1- x)^{n- k}\end{aligned} \] and \(\phi_ n= n+ \alpha_ n\), where \(\alpha_ n\geq 0\) and \(\lim_{n\to\infty} n^{-{1\over 2}} \alpha_ n= 0\). In this paper, the author studies the rates of convergence of the positive linear operators \(L_ n\) and \(L^*_ n\). The motivation for this work was the paper by \textit{R. Bojanic} and \textit{M. Vuillemeier} [J. Approximation Theory 31, 67-79 (1981; Zbl 0494.42003)].