A new type of Szász–Mirakjan operators based on q-integers

Abstract

Abstract In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Φ n , q ( t m ; x ) $\Phi _{n,q}(t^{m};x)$ for m = 0 , 1 , 2 $m=0,1,2$ and the central moments Φ n , q ( ( t − x ) m ; x ) $\Phi _{n,q}((t-x)^{m};x)$ for m = 1 , 2 $m=1,2$ . We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Keywords

• q-calculus
• q-Bernstein-polynomials
• q-Szász-operators
• Moments
• Modulus of continuity