Pointwise estimates for monotone polynomial approximation

We prove that iff is increasing on [−1,1], then for eachn=1,2,... there is an increasing algebraic polynomialPn of degreen such that |f(x)−Pn(x)|≤cω2(f,√1−x2/n), whereω2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.