Monotone and comonotone polynomial approximation revisited

Using a suitable Peetre functional that weighs differently the behaviour of the function in the middle of the interval and near the endpoints, we obtain estimates of the Jackson type on the rate of monotone polynomial approximation to a monotone continuous function. These estimates involve the second modulus of smoothness related to the Peetre functional. Then we apply these estimates to get estimates on the degree of comonotone polynomial approximation of a piecewise monotone function and on the degree of copositive polynomial approximation of a continuous function that changes sign finitely often in the interval.