SOME POSITIVE RESULTS AND COUNTEREXAMPLES IN COMONOTONE APPROXIMATION

Let f be a continuous function on [-1, 1], which changes its monotonic ity finitely many times in the interval, say s times. We discuss the v alidity of Jackson-type estimates for the approximation of f by algebr aic polynomials that are comonotone with it. While we prove the validi ty of the Jackson-type estimate involving the Ditzian-Totik modulus of continuity and a constant which depends only on s, we show by counter examples that in many cases this is not so, even for functions which p ossess locally absolutely continuous derivatives. These counterexample s are given when there are certain relations between s, the number of changes of monotonicity, and r, the number of derivatives. For other c ases we do have some Jackson-type estimates and another paper will be devoted to that. (C) 1997 Academic Press.