Strong converse inequality for the Bernstein-Durrmeyer operator
The Bernstein-Durrmeyer operator is given by \[ M_ n(f,x)= (n+ 1) \sum^ n_{k=0} P_{n,k}(x) \int^ 1_ 0 P_{n,k}(y)f(y)dy, \] where \(P_{n,k}(x)={n\brack k} x^ k(1- x)^{n-k}\) \((0\leq x\leq 1)\). An equivalence relation between the rate of approximation of Bernstein- Durrmeyer polynomials and an appropriate \(K\)-functional is established. The results obtained are stronger than those known for Bernstein polynomials. For earlier work on this theme, see \textit{M. M. Derrienic} [J. Approximation Theory 31,325-343 (1981; Zbl 0475.41025)].
scientific article; zbMATH DE number 465060