On the rate of convergence of Bernstein polynomials of functions of bounded variation (Q789635)

Let f be a real valued function of bounded variation on [0,1]. Define \(g_ x\) by \(g_ x(t)=f(t)-f(x+)\), \(x<t\leq 1\), \(g_ x(t)=0\), \(t=x\), \(g_ x(t)=f(t)-f(x-), 0\leq t<x\) for a fixed x in (0,1). Let \(B_ n(f)\) be the n-th Bernstein polynomial of f. The author proves for every x in (0,1) and every \(n\geq(3/x(1-x))^ 8\) the estimate \[ | B_ n(f,x)- (1/2)(f(x+)+f(x-))| \leq \frac{3(x(1-x))^{- 1}}{n}\sum^{n}_{k=1}V_{x-x/\sqrt{k}}\!\!\!\!\!\!\!^{x+(1- x)/\sqrt{k}}(g_ x)+ \] \[ +\frac{18(x(1-x))^{-5/2}}{n^{1/6}}| f(x+)-f(x-)|, \] where \(V^ b\!_ a(g_ x)\) denotes the total variation of \(g_ x\) on [a,b].