On the composition and decomposition of positive linear operators (VII)

In the present paper we study compositions of piecewise linear interpolation operator S?n and Beta-type B?n, namely An:= ?B?n Gn := B?n ? S?n. Voronovskaya type theorems for operators An are proved, substantially improving some corresponding known results. The rate convergence iterates is considered. Some estimates differences between An, Gn, Bn S?n, respectively, given. Also, behaviour on subspace C[0,1] consisting all polygonal functions with nodes {0, 1/2,..., n-1/n,1}. Finally, propose to readers a conjecture concerning eigenvalues Gn. If true, this would emphasize new strong relationship classical Bernstein Bn.