Some positive results and counterexamples in comonotone approximation (Q1356800)

Let \(f\) be a continuous function on \([-1,1]\), which changes its monotonicity \(s\) times in the interval. This paper proves the validity of the Jackson-type estimates for the approximation of \(f\) by algebraic polynomials that are comonotone with it involving the Ditzian-Totik modules of continuity and a constant that depends only on \(s\). This paper also shows by counterexamples that in many cases this is not so, even for functions which possess locally absolutely continuous derivatives.