On Szász–Mirakyan Operators Preserving $$\varvec{e^{2ax}},$$ e 2 a x , $$\varvec{a>0}$$ a > 0

Abstract

A modification of Szasz–Mirakyan operators is presented that reproduces the functions 1 and $$e^{2ax}$$ , $$a>0$$ fixed. We prove uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya-type theorem. A comparison with the classical Szasz–Mirakyan operators is given. Some shape preservation properties of the new operators are discussed as well. Using a natural transformation, we also present a uniform error estimate for the operators in terms of the first- and second-order moduli of smoothness.