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Approximation properties of Szasz type operators based on Charlier polynomials

In the present paper, we study some approximation properties of the Szasz type operators involving Charlier polynomials introduced by Varma and Tasdelen in 2012. First, we establish approximation in a Lipschitz type space and weighted approximation theorems for these operators. | Turk J Math (2015) 39: 990 – 1003 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1502-80 Research Article Approximation properties of Sz´ asz type operators based on Charlier polynomials Arun KAJLA∗, Purshottam Narain AGRAWAL Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India Received: 28.02.2015 • Accepted/Published Online: 12.07.2015 • Printed: 30.11.2015 Abstract: In the present paper, we study some approximation properties of the Sz´ asz type operators involving Charlier polynomials introduced by Varma and Ta¸sdelen in 2012. First, we establish approximation in a Lipschitz type space and weighted approximation theorems for these operators. Then we obtain the error in the approximation of functions having derivatives of bounded variation. Key words: Sz´ asz operator, Charlier polynomials, modulus of continuity, bounded variation 1. Introduction In 1950, Sz´asz [20] introduced the following linear positive operators: Sn (f ; x) = e −nx ( ) ∞ ∑ (nx)k k f , k! n (1.1) k=0 where x ∈ [0, ∞) and f (x) is a continuous function on [0, ∞) whenever the above sum converges uniformly. Many researchers have studied approximation properties of these operators and modified Sz´asz operators by involving different types of polynomials. Jakimovski and Leviatan [13] defined a generalization of Sz´asz operators including the Appell polynomials and gave the approximation properties of these operators. In [21], Varma et al. considered the generalization of Sz´asz operators involving Brenke type polynomials and studied convergence properties by using the Korovkin type theorem and the order of convergence with the help of classical method. Recently, Altomare et al. [3] introduced a new kind of generalization of Sz´asz–Mirkajan–Kantorovich operators and obtained the rate of convergence by means of suitable moduli of smoothness. Several researchers also defined different types of .