On the Cheney and Sharma operator
The authors consider the double indexed Bernstein power series operator \(P_{a,t}\) introduced by \textit{E. W. Cheney} and \textit{A. Sharma} [Can. J. Math. 16, 241-252 (1964; Zbl 0128.29001)] and related to the classical Laguerre polynomials. Sharp estimates of the first two moments are obtained. These enable the authors to prove that, when acting on a continuous function \(f\), the convergence of \(P_{a,t}f\) to \(f\) is uniform on the whole interval \([0,1]\). Further, it is shown that the rates of convergence depend on the way in which \(t/a\) tends to \(0\). It is also shown that \(P_{a,t}\) preserves monotonicity and global smoothness. Finally, the authors discuss the monotonicity properties of \(P_{a,t}\) with respect to both parameters. A probabilistic approach is adopted throughout.
scientific article; zbMATH DE number 908597