Statistical approximation by Meyer‐König and Zeller operators of finite type based on the q‐integers

Gavrea and Trif [I. Gavrea, T. Trif, The rate of convergence by certain new Meyer‐König and Zeller operators of finite type, Rend. Circ. Mat. Palermo (2) Suppl. 76 (2005) 375–394] introduced a sequence $\left(L_n\right)$ of Meyer‐König and Zeller operators ‘of finite type’ and investigated the rate of convergence of these operators for continuous functions. In the present paper we generalize these operators to the framework of $q$ ‐calculus. By deriving a sharp estimate of the second moment, we establish a Bohman–Korovkin type approximation theorem for the new $L_{n,q}$ ‐operators via $A$ ‐statistical convergence. We also compute the rate of $A$ ‐statistical convergence of the $L_{n,q}$ ‐operators in terms of Peetre’s functional.