Some positive results and counterexamples in comonotone approximation

Letfbe a continuous function on [-1, 1], which changes its monotonicity finitely many times in the interval, saystimes. We discuss the validity of Jackson-type estimates for the approximation offby algebraic polynomials that are comonotone with it. While we prove the validity of the Jackson-type estimate involving the Ditzian-Totik modulus of continuity and a constant which depends only ons, we show by counterexamples that in many cases this is not so, even for functions which possess locally absolutely continuous derivatives. These counterexamples are given when there are certain relations betweens, the number of changes of monotonicity, andr, the number of derivatives. For other cases we do have some Jackson-type estimates and another paper will be devoted to that.