On the rate of convergence of the integrated Meyer-König and Zeller operators for functions of bounded variation
The author considers for \(f\in L_ 1[0,1]\) the integrated Meyer-König and Zeller operator \(\hat M_ n(f,x)=\sum^{\infty}_{k=0}\hat M_{nk}(x)\int_{I_ k}f(t)dt\), converging for \(0\leq x<1\), with \(I_ k=[\frac{k}{n+k},\frac{k+1}{n+k+1}]\), \(k\in {\mathbb{N}}\), \(\hat M_{nk}(x)=(n+1)\left( \begin{matrix} n+k+1\\ k\end{matrix} \right)x^ k(1-x)^ n\). Using some results of probability theory (a version of the central limit theorem, the author gives an estimate for the rate of convergence of \(\hat M_ n(f,x)\) for functions of bounded variation. He also shows that the estimates are essentially best possible. The main theorem reads: Let f be a function of bounded variation on [0,1] and let \(V^ b_ a(g_ x)\) be the total variation of \(g_ x\) on [a,b]. Then, for every \(x\in (0,1)\) and n sufficiently large, we have \[ | \hat M_ n(f,x)- (f(x+)+f(x-))| \leq \frac{5}{nx}\sum^{n}_{k=1}\bigvee^{x+(1- x)/\sqrt{k}}_{x-x/\sqrt{k}}(g_ x)+ \] \[ \frac{49}{\sqrt{n}x^{3/2}}| f(x+)-f(x-)|, \] where \(g_ x(t)=f(t)-f(x+)\) for \(x<t\leq 1\), \(=0\) for \(t=x\), and \(=f(t)-f(x-)\) for \(0\leq t<x\).
scientific article; zbMATH DE number 4109299