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

It 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