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We introduce new moduli of smoothness for functions f ∈ L p [−1, 1] ∩ C r −1 (−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that ϕ r f (r ) These moduli are equivalent to certain weighted Ditzian-Totik (DT) moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in L p [−1, 1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.Communicated by Kamen Ivanov.Part of this work was done while the first two authors were at

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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 | 3 . On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allows | 4 estimates. Academic Press