On Szasz-Mirakyan Operators Preserving e(2ax), 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.