Pointwise estimates for monotone polynomial approximation
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We prove that if f is increasing on [-1,1], then for each n=1,2,... there is an increasing algebraic polynomial Pn of degree n such that |f(x)-Pn(x)|c2(f,1-x2/n), where 2 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. 1985 Springer-Verlag New York Inc.