Nearly comonotone approximation

We discuss the degree of approximation by polynomials of a function f that is piecewise monotone in [ - 1, 1]. We would like to approximate f by polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show here that in that case the polynomials can achieve the rate of omega(3). On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allows omega(4) estimates. (C) 1998 Academic Press.