Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.