On equivalence of moduli of smoothness of polynomials in Lp, 0<p⩽∞

AbstractIt is well known that ωr(f,t)p⩽tωr-1(f′,t)p⩽t2ωr-2(f″,t)p⩽⋯ for functions f∈Wpr, 1⩽p⩽∞. For general functions f∈Lp, 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=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the Lp space for 0