*Article* **Approximation by Generalized Lupa¸s Operators Based on** *q***-Integers**

#### **Mohd Qasim 1, M. Mursaleen 2,3,4,\*, Asif Khan 3 and Zaheer Abbas 1**


Received: 6 December 2019; Accepted: 27 December 2019; Published: 2 January 2020

**Abstract:** The purpose of this paper is to introduce *q*-analogues of generalized Lupa¸s operators, whose construction depends on a continuously differentiable, increasing, and unbounded function *ρ*. Depending on the selection of *q*, these operators provide more flexibility in approximation and the convergence is at least as fast as the generalized Lupa¸s operators, while retaining their approximation properties. For these operators, we give weighted approximations, Voronovskaja-type theorems, and quantitative estimates for the local approximation.

**Keywords:** generalized Lupas. operators; *q* analogue; Korovkin's type theorem; convergence theorems; Voronovskaya type theorem
