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 En(f ) ≤ n−α , n ≥ 1. We ask whether the degree of comonotone approximation is necessarily ≤ c(α, s)n−α , n ≥ 1, and if not, what can be said. It turns out that for each s ≥ 1, there is an exceptional set As of α’s for which the above estimate cannot be achieved.