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A note on integral modification of the Meyer-König and Zeller operators. (English) Zbl 1023.41014
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 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, \[ \begin{split}\biggl|\widehat M_n(f,x)- {1\over 2}\bigl\{f(x^+) +f(x^-)\bigr\} \biggr|\leq{7\over nx} \sum^n_{k=1} V^{x+(1-x)/ \sqrt k}_{x-x/ \sqrt k}(g_x)+ \left(3+{1 \over \sqrt 8e}\right) {1\over\sqrt nx^{3/2}}\\ \bigl|f(x^+)-f(x^-)\bigr |, \end{split} \] where \(V^b_a(g_x)\) is the total variation of \(g_x\) on \([a,b]\) and \[ g_x(t)= \begin{cases} f(t)-f(x^-), \quad & 0\leq <x\\ 0,\quad & t=x\\ f(t)-f(x^+), \quad & x<t\leq 1.\end{cases}. \]
MSC:
41A30 Approximation by other special function classes
41A36 Approximation by positive operators
41A25 Rate of convergence, degree of approximation
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