On equivalence of moduli of smoothness of polynomials in L-P, 0 < p <= infinity

摘要

It is well known that omega(r) (f, t) p <= t omega(r-1) (f(t), t) p <= t(2) omega(r-2) (f", t) p <= . . . for functions f epsilon W-p(r), 1 <= p <= infinity . For general functions f epsilon L-p, it does not hold for 0 < p < 1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p = infinity. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L-p space for 0 < p <= infinity. We then show its analogues for the Ditzian-Totik modulus of smoothness (omega(phi)(r) (f, t) p and the weighted Ditzian-Totik modulus of smoothness omega(phi)(r)(f, t)(omega,p) for polynomials with phi(x) = root 1 - x(2). (c) 2005 Elsevier Inc. All rights reserved.