On Equivalence of Moduli of Smoothness Original Research Article

It 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.