nu m b er 11 36 WEIGHTED D - T MODULI REVISITED AND APPLIED

We introduce weighted moduli of smoothness for functions f ∈ Lp[−1, 1]∩Cr−1(−1, 1), r ≥ 1, that have an (r−1)st absolutely continuous derivative in (−1, 1) and such that φf (r) is in Lp[−1, 1], where φ(x) = (1−x). These moduli are equivalent to certain weighted D-T 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 D-T moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in Lp[−1, 1] (no weight), by means of algebraic polynomials. We also have some inverse theorems that yield characterization of the behavior of the derivatives of the function by means of its degree of approximation. 1. Definitions Let ‖ · ‖p := ‖ · ‖Lp[−1,1], 1 ≤ p <∞, and φ(x) = √ 1− x2. For k ∈ N0, let ∆h(f, x, J) :=  k ∑ i=0 ( k i ) (−1)k−if(x+ (i− k/2)h), if x± kh/2 ∈ J , 0, otherwise, be the kth symmetric difference, and let ∆h(f, x) := ∆ k h(f, x, [−1, 1]). Definition 1.1. Let 1 ≤ p <∞ and r ∈ N0. Then for r ≥ 1, let Bp := { f | f (r−1) ∈ ACloc(−1, 1) and ‖f φ‖p < +∞}, and set Bp := Lp[−1, 1]. For f ∈ Bp define ω k,r(f , t)p := sup 0<h≤t ∥∥Wkh(·)∆khφ(·)(f , ·)∥∥p , 2010 Mathematics Subject Classification. 41A10, 41A17, 41A25.