Best direct and converse results for Lagrange-type operators

The aim of this paper is to give direct and converse theorems for the approximation by using a discretely defined method L _n ^* (see D. H. Mache 10], which is a modification of the Lagrange operator. Furthermore we obtain with a matrix construction technique (see M. D. Ye and D. X. Zhou 11]) a Lagrange-type operator ℒ_n, for which we get a characterization for Lipschitz functions by the approximation rate of these methods. The second author is supported by the Alexander von Humboldt-Stiftung.