Some Positive Results and Counterexamples in Comonotone Approximation, II

AbstractLetfbe a continuous function on [−1, 1], which changes its monotonicity finitely many times in the interval, saystimes. In the first part of this paper we have discussed the validity of Jackson type estimates for the approximation offby algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estimate involving the Ditzian–Totik (first) modulus of continuity and a constant which depends only ons, and we have shown by counterexamples that in many cases the Jackson estimates involving the DT-moduli do not hold when there are certain relations betweens, the number of changes of monotonicity, andr, the number of derivatives of the approximated function. Here we deal with all other cases and we obtain Jackson type estimates involving modified DT-moduli. We also provide counterexamples to complete the picture. Our technique for the positive results involves a two-tier approach. We first approximate the given function by comonotone piecewise polynomials which yield good approximation and then we replace the latter by polynomials