Applications of the Shorgin Identity to Bernstein Type Operators

Abstract

By establishing an identity between a sequence of Bernstein-type operators and a sequence of Szász–Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szàsz–Mirakyan operators. As one application of this identity, we prove that whenever the parameters are conveniently chosen, if $f\in C[0,\infty )$ satisfies a growth condition of the form $|f(t)|\le C e^{\alpha t}(C,\alpha \in \mathbb {R}^+)$, then the classical Bernstein operators $B_{mn}(f(nu),x/n)$ converge to the Szàsz–Mirakyan operator $S_m(f,x)$. This convergence generalizes the classical result of De la Cal and Liquin to unbounded functions; moreover, the rth derivative of $B_{mn}(f(nu),x/n)$ converges to the rth derivative of $S_m(f,x)$. As another application of this identity, we derive Voronowskaja type result for the general Lototsky–Bernstein operators.