Optimal Estimation Rules for Functions of High Smoothness

Let I denote a linear functional defined on the space of 2-periodiccontinuous functions. It is an often used technique to approximateI[f] by I?intpol [f], where intpol [f] means a trigonometricinterpolation polynomial of f. The aim of this paper is tomakeclear that this technique is almost optimal for functions ofhigh smoothness. As a by-product we obtain a new proof of aresult of Schoenberg (1972) and v. Golitschek (1972) concerninglimits of interpolating spline functions.