Constants in comonotone polynomial approximation - A survey

We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of continuous and r-times continuously differentiable functions which change their monotonicity finitely many times in a finite interval, say [-1, 1], with special attention to the constants involved in the estimates. We describe four possibilities ranging from the existence of estimates involving absolute constants and which are valid for polynomials of all degrees except the very few first ones, through estimates where either the constants or the degrees of the polynomials depend on the location of the points of change of monotonicity, then through estimates where either the constants or the degrees of the polynomials depend on the specific function, and finally cases where there are no Jackson type estimates possible.