Interpolatory estimates in monotone piecewise polynomial approximation

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Given a monotone function f∈Cr[−1,1], r≥1, we obtain pointwise estimates for its monotone approximation by piecewise polynomials involving the second order modulus of smoothness of f(r). These estimates are interpolatory estimates, namely, the piecewise polynomials interpolate the function at the endpoints of the interval. However, they are valid only for n≥N(f,r). We also show that such estimates are in general invalid with N independent of f.