Some positive results and counterexamples in comonotone approximation. II (Q1806140)

For part I see the authors in ibid. 89, No. 2, 195-206 (1997; Zbl 0870.41016). Let \(f\) be a continuous function on \([-1,1]\), which chnges its monotonicity \(s<+\infty\) times. This paper proves the validity of a Jackson type estimate involving the Ditzian-Totik (first) modulus of continuity and a constant which depends only on \(s\) for the approximation of \(f\) by algebraic polynomials that are comonotone with it. But this paper also shows by counterexamples that in many cases the Jackson estimate involving the Ditzian-Totik moduli do not hold when there are certain relations between \(s\) and \(r\), the number of derivatives of the approximated function.