Rate of Pointwise Convergence of Bernstein Polynomials for Some Absolutely Continuous Functions

Given any absolutely continuous function f having the derivative f X equivalent to a function w bounded on w0, 1x, the rate of convergence of the Bernstein polynomials Bn f x. is estimated at those points xg 0, 1. at which the one-sided limits w xq., w xy. exist. In the main result the variations of positive integer orders are used. In the special case when w gBVw0, 1x the improved version of the Bojanic and Cheng estimate w2x is obtained.