On Equivalence of Moduli of Smoothness

AbstractIt is known that iff∈Wkp, thenωm(f,t)p⩽tωm−1(f′,t)p⩽…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusωm(S,t)p∼tωm−1(S′,t)p∼t2ωm−2(S″,t)p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper