Approximation by modified Szász-Durrmeyer operators

Abstract

The main goal of this paper is to introduce Durrmeyer modifications for the generalized Szász–Mirakyan operators defined in (Aral et al., in Results Math 65:441–452, 2014). The construction of the new operators is based on a function $\rho$ which is continuously differentiable $\infty$ times on $\left[ 0,\infty \right) ,$ such that $\rho \left( 0\right) =0$ and $\inf _{x\in \left[ 0,\infty \right) }\rho ^{\prime }\left( x\right) \ge 1.$ Involving the weighted modulus of continuity constructed using the function $\rho$, approximation properties of the operators are explored: uniform convergence over unbounded intervals is established and a quantitative Voronovskaya theorem is given. Moreover, we obtain direct approximation properties of the operators in terms of the moduli of smoothness. Our results show that the new operators are sensitive to the rate of convergence to f,  depending on the selection of $\rho .$ For the particular case $\rho \left( x\right) =x$, the previous results for classical Szász-Durrmeyer operators are captured.