A generalization of Szász-type operators which preserves constant and quadratic test functions

Abstract: In the present article, we introduced a new form of Szász-type operators which preserves test functions e 0 and e 2 (e i (t) = t i , i = 0, 2). By these sequence of positive linear operators, we gave rate of convergence and better error estimation by means of modulus of continuity. Moreover, we have discussed order of approximation with the help of local results. In the last, weighted Korovkin theorem is established.


Introduction
For f ∈ C(0, +∞) and x ∈ (0, +∞), Szász (1950) defined a sequence of positive linear operators as follows where p n,k (x) = e −nx (nx) k k! , n ∈ ℕ. These operators play important role in approximation theory. In this paper, Szász showed the manner in which the operators S n (f ;x) tend to f(x). Various well-known positive linear operators L n preserve the constant as well as the linear functions i.e. L n (e 0 ;x) = e 0 (x) and L n (e 1 ;x) = e 1 (x) for the test functions e i (x) = x i (i = 0, 1). But, these operators do not preserve e 2 (x) and it is rather difficult to approach e 2 (x) for large value of n (see also Rao & Wafi, 2015;Wafi & Rao, 2016). In King (2003)

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The approximation of functions by positive linear operators plays a significant role in the areas of numerical analysis, computer-aided geometric design (CAGD), solutions of differential equations, etc. In this paper, we modified Szász type operators based on Charlier polynomials using King's method to obtain better approximation results. Moreover, we discussed basic convergence theorem in terms of classical modulus of continuity, investigated pointwise approximation theorems and weighted approximation theorem. e 2 (x) i.e. L n (e 2 ;x) = e 2 (x) and provided the better error estimation. Several authors used this powerful tool to different type of positive linear operators and discussed the better error estimation for instance (Ali Özarslan & Duman, 2009;Deo & Bhardwaj, 2015;Duman & Ozarslan, 2007). Recently, Varma and Tasdelen (2012) gave a generalization of well-known Szász-Mirakjan operators using Charlier polynomials (Ismail, 2005) having the generating function of the form: and the explicit representation where ( ) k is the Pochhammer's symbol given by For a > 0 and u ≤ 0, Charlier polynomials are positive. Using these polynomials, they (Varma & Taşdelen, 2012) defined the Szász-type operators as follows: where a > 1 and x ≥ 0. Now, we introduce a new sequence of Szász-type operators which preserves constant and quadratic test functions and provides better estimates. Let T n,a :C[0, +∞) → C[0, +∞). (ii) for r n,a (x) = x, a → +∞ and x − 1 n in place of x, these operators tend to classical Szász operators defined by (1.1).
In this paper, we have discussed rate of convergence, local approximation results and Korovkintype approximation theorem in polynomial weighted space and obtained better estimates for the operators (1.4).

Basic estimates
Lemma 2.1 For the operators T n,a defined by (1.4), we have T n,a (1;r n,a (x)) = 1, T n,a (t;r n,a (x)) = Proof Using t = 1, u = −(a − 1)nr n,a (x) in (1.2) and by simple differentiation, we get Using these equalities and operators (1.4), we can easily prove Lemma 2.1. ✷  x.

Remark 3.1 For the Szász-type operators L n given by (1.3), and for every
A ∈ R and M ∈ (0, ∞)} Here, we show that our operators T n,a has the better approximation than the operators L n .

Local approximation results
In this section, we deal with the order of approximation locally in C B [0, ∞) (space of real-valued continuous and bounded functions f defined on [0, ∞)) with the norm ‖f ‖ = sup 0≤x<∞ �f (x)�. Then, for any f ∈ C B [0, ∞) and > 0, Peeter's K-functional is defined as DeVore and Lorentz (1993, p. 177, Theorem 2.4), there exits an absolute constant C > 0 such that where 2 (f ; ) is the second-order modulus of continuity is defined as x ;r n,a (x) + T n,a x ;r n,a (x) 2 .
Proof First, we consider the auxiliary operators as follows where n,a (x) = T n,a x ;r n,a (x) + x. By the Equation (4.1), we get For any g ∈ C 2 B [0, ∞) and by the Taylor's theorem, we have Applying auxiliary operators defined by (4.1) in Equation (4.3), we get where M is a constant and 0 < ≤ 1.

Weighted Korovkin-type theorem
In this section, we introduce T n,a in polynomial weighted spaces of continuous and unbounded functions defined on [0, ∞). In Gadzhiev (1976) gave the weighted Korovkin-type theorems. Here we recall some symbols and notions from Gadziev (1976). Let (x) = 1 + x 2 , −∞ <