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 Szasz–Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szasz–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 Szasz–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.