Parametric generalization of Baskakov operators

Abstract

Herein we propose a non-negative real parametric generalization of the Baskakov operators and call them as $\alpha$-Baskakov operators. We show that $\alpha$-Baskakov operators can be expressed in terms of divided differences. Then, we obtain $n$th order derivative of $\alpha$-Baskakov operators in order to obtain its new representation as powers of independent variable $x$. In addition, we obtain Korovkin’s type approximation properties of $\alpha$-Baskakov operators. Moreover, by using the modulus of continuity, we obtain the rate of convergence. Numerical results presented show that depending on the value of the parameter $\alpha$, an approximation to a function improves compared to the classical Baskakov operators.