On the rate of pointwise convergence of modified Baskakov type operators (Q1370577)

Let \(I=[0,\infty)\) and let \(M_0(I)\) be the class of measurable complex-valued locally bounded functions on \(I\). For \(f\in M_0(I)\) the modified Baskakov type operators \[ B_n(f;x)= \sum_{k=0}^\infty p_{n,k}(x) \int_0^\infty f(t)b_{n,k}(t)dt, \qquad n\in\mathbb{N}, \quad x\in I, \] where \(p_{n,k}(x)= {{n+k-1}\choose k}x^k(1+x)^{-n-k}\), \(b_{n,k}(t)= t^kB^{-1}(k+1,n) (1+t)^{-n-k-1}\), \(B(k+1,n)= k!(n-1)! ((n+k)!)^{-1}\). In this paper the authors present general quantitative inequalities concerning the rate of pointwise convergence of \(B_n(f;x)\) for some functions \(f\in M_0(I)\) at those points \(x\in (0,\infty)\) at which the one-sided limits \(f(x\pm)\) exists. Our main estimates will be expressed in terms of the modulus of variation of the auxiliary function \(g_x\) defined by \(g_x(t)= f(t)- f(x-)\) for \(0\leq t<x\), \(g_x(x)=0\) and \(g_x(t)= f(t)-f(x+)\) for \(x<t<\infty\).