A note on integral modification of the Meyer-König and Zeller operators (Q1811962)

The Meyer-König and Zeller operators \(M_n (n\in\mathbb{N})\) are defined by \[ M_n(f,x)= \sum^\infty_{k=0} p_{n,k}(x)f ({k\over n+k}),\;x\in [0,1] \tag{1} \] and \[ p_{n,k}(x)= {n+k-1\choose k} x^k(1-x)^n.\tag{2} \] The integral modifications of operators (1) were given by S. S. Guo (see \textit{S. S. Guo} [Approximation Theory Appl. 4, 9-18; Zbl 0686.41016)]) as follows \[ \widehat M_n(f,x) =\sum^\infty_{k=1} p_{n,k+1}(x) {(n+k-2) (n+k-3)\over n-2}\int^1_0 p_{n-2,k-1} (t)f(t)dt\tag{3} \] where \(p_{n,k}(x)\) is defined in (2). In their note, the authors give an improved estimate for the rate of convergence of functions of bounded variation for the operators (3). This result is the following theorem. Let \(f\) be a function of bounded variation on \([0,1]\). Then, for every \(x\in (0,1)\) and \(n\) sufficiently large, \[ \begin{multlined}\biggl|\widehat M_n(f,x)- {1\over 2}\bigl\{f(x^+) +f(x^-)\bigr\} \biggr|\leq{7\over nx} \sum^n_{k=1} V^{x+(1-x)/ \sqrt k}_{x-x/ \sqrt k}(g_x)+ \left(3+{1 \over \sqrt 8e}\right) {1\over\sqrt nx^{3/2}}\\ \bigl|f(x^+)-f(x^-)\bigr |, \end{multlined} \] where \(V^b_a(g_x)\) is the total variation of \(g_x\) on \([a,b]\) and \[ g_x(t)= \begin{cases} f(t)-f(x^-), \quad & 0\leq <x\\ 0,\quad & t=x\\ f(t)-f(x^+), \quad & x<t\leq 1.\end{cases}. \]