Positive linear operators with equidistant nodes

Abstract

In the present paper, the approximation power of positive linear operators with equidistant nodes is investigated. New pointwise estimates are given in terms of first and second order moduli of continuity, showing that for positive operators having uniform Jackson orders of approximation one may expect interpolation at the endopoints. In particular, the first solution to Butzer's problem with equidistant nodes is given. A negative result yields an explanation why Bernstein operators and some of their modifications are optimal in a certain sense.