Pointwise Estimates for Monotone Polynomial Approximation

We prove that if f is increasing on [ 1,1], then for each n = 1, 2 . . . . . there is an increasing algebraic polynomial P. of degree n such that {f(x) P.(x){ < cw2( f, V/I x 2 /n) , where w2 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.