Some Positive Results and Counterexamples in Comonotone Approximation Ii

Let f be a continuous function on ?1; 1], which changes its monotonicity nitely many times in the interval, say s times. In the rst part of this paper we have discussed the validity of Jackson type estimates for the approximation of f by algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estimate involving the Ditzian {Totik ((rst) modulus of continuity and a constant which depends only on s, and we have shown by counterexamples that in many cases the Jackson estimates involving the D{T moduli, do not hold when there are certain relations between s, the number of changes of monotonicity and r, the number of derivatives of the approximated function. Here we deal with all other cases and we obtain Jackson type estimates involving modiied D{T moduli. We also provide counterexamples to complete the picture. Our technique for the positive results involves a two-tier approach. We rst approximate the given function by comonotone piecewise polynomials which yield good approximation and then we replace the latter by polynomials x1. Introduction Let f 2 C ?1; 1] change monotonicity nitely many times, say s 1, in the interval, and we wish to approximate f by polynomials p n 2 P n , the space of polynomials of degree not exceeding n, which are comonotone with f. To be speciic, let s 1 and let Y s be the set of all collections Y := fy i g s i=1 of points, ?1 < y s < ::: < y 1 < 1. For Y 2 Y s we set (x; Y) := s Y i=1 (x ? y i) ; and denote by (1) (Y) the set of functions f 2 C ?1; 1], which change monotonicity at the points y i , and which are nondecreasing in (y 1 ; 1), that is, f is nondecreasing in