Lower bounds for the approximation with variation-diminishing splines

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval [0,1][0,1] in terms of classical moduli of smoothness depending on the degree of the spline basis. For this purpose we use a functional analysis framework in order to characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates.