Approximation by modified Kantorovich–Szász type operators involving Charlier polynomials

In this paper, we give some direct approximation results by modified Kantorovich–Szász type operators involving Charlier polynomials. Further, approximation results are also developed in polynomial weighted spaces. Moreover, for the functions of bounded variation, approximation results are proved. Finally, some graphical examples are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions.

Motivated by the work done in [9], we define the Kantorovich generalization [10] of (1.2) as follows: where μ n and ν n are sequences of positive numbers which are increasing and unbounded such that If we take μ n = ν n = n, we will have the operators defined in [9].

Auxiliary results
We first present some auxiliary results. (1 + μ n y), From the above equalities, the claims of the lemma can be obtained.
given by (1.3), we have the following equalities:

Local approximation results
In what follows, let Q (μ n ,ν n ) n,b (ty; y) = χ μ n ,ν n (y) and Q (μ n ,ν n ) n,b ((ty) 2 ; y) = ξ μ n ,ν n (y). We will now give two theorems on the uniform convergence and the order of approximation.
The proof of the theorem is established by taking advantage of the above uniform convergence in each compact subset of [0, ∞) and the famous Korovkin's theorem.
Suppose f ∈C[0, ∞), i.e., f belongs to the space of uniformly continuous functions on [0, ∞). If δ > 0, then the modulus of continuity ω(f , δ) is defined by Proof From (1.3) and the property of modulus of continuity, the left-hand side of (3.1) leads to Using Cauchy-Schwarz inequality for the integral, we get In the above sum, we apply Cauchy-Schwarz inequality, and then in view of Lemma 2.1, where, taking δ = 1 ν n , we get (3.1).
Let a 1 , a 2 > 0 be fixed. We now consider the following space of Lipschitz type (see [18]): where M is a positive constant and 0 < r ≤ 1.
We denote the space of all functions h on [0, 1) which are real-valued, uniformly continuous, as well as bounded byC B [0, ∞) and endow it with the norm h ∞ = sup y∈[0,1) |h(y)|. Further, we obtain a local direct estimate for the operators (1.3), using the Lipschitz maximal function of order r introduced by Lenze [13] as: where y ∈ [0, 1) and r ∈ (0, 1]. Proof By equation (3.4), Applying on both sides of the above inequality, then using Lemma 2.1, as well as Hölder's inequality with p = 2/r, q = 2/(2r), we obtain Thus, we have our desired result.
The Peetre's K -functional is given by holds for all δ > 0, where ω 2 is the second-order modulus of smoothness of g ∈C B [0, ∞), which is defined by .
Proof For f ∈C B [0, ∞), we define the auxiliary operator as follows: After taking the absolute value of both sides, , using Taylor's theorem, we can write Applying operatorQ (μ n ,ν n ) n,b to the above equation, we get Now taking the absolute value of both sides, we obtain Therefore, by using the norm on g, we have Now, using the definition of auxiliary operators (3.5), we get Combining (3.6) and (3.7) with the above equation, we get and after taking the infimum on the right-hand side over all g ∈C 2 B , we have This completes the proof of the theorem. .
Now, using the well-known property of the modulus of continuity for δ > 0 and f ∈ Therefore, from (3.8) and the above equation, we have After applying the Cauchy-Schwarz inequality, we get (1; y) Choosing δ = δ n (y), we get our desired result.
For f ∈C B [0, ∞), the Ditzian-Totik modulus of smoothness [4] of the first order is given by and an appropriate Peetre's K -functional is defined by Proof Let ϕ(y) = √ y, then by Taylor's theorem, for any g ∈ W ϕ [0, ∞), we get therefore, Using Lemma 2.1 and the above equation, for any g ∈ W ϕ [0, ∞), we get |t -y|; y .
Applying the Cauchy-Schwarz inequality yields Taking infimum on the right-hand side over all g ∈ W ϕ [0, ∞), we get which leads to the required result with the help of the relation between Peetre's Kfunctional and Ditzian-Totik modulus of smoothness as given by the relation (3.9).  (1 + y 2 ) 1+r .
In order to study the order of convergence of the operators Q (μ n ,ν n ) n,b (f ; y) for the functions having a derivative of bounded variation, we rewrite the operator (1.3) as follows: (4.9) where W (t, y) is a kernel given by χ I (t) being the characteristic function of I = [ l ν n , l+1 ν n ].

Lemma 4.2 Let for all x > 0 and sufficiently large n,
Proof Using Lemma 2.1 and the definition of the kernel, we get Hence, we have In the same fashion, we can prove the other inequality, therefore, we omit the details.
Let Let Proof By (4.11), we obtain where Now using Lemma 2.1, equations (4.9) and (4.12), we get Since t y δ y (u) du = 0, we have Now, we break the second term on the right-hand side of the above equation as follows: where Taking the absolute value on both sides of (4.13), we have |t -y|; y .
After applying the Cauchy-Schwarz inequality, we obtain Now applying Lemma 4.2 and integration by parts, I 1 can be written as On taking the absolute value of I 1 , we have f y (t) λ μ n ,ν n (t, y) dt + y y-y/ √ n f y (t) λ μ n ,ν n (t, y) dt = K 1 + K 2 , say.
Hence, we get y+y/k y f y Now, from (4.14)-(4.16), we obtain Also, a direct comparison between the convergence of the old operator applied to f (when μ n = ν n = n discussed in [9]) (blue) and the new operator (red) defined in Eq. (1.3) towards f (x) (cyan) is shown in Figs. 7-9, respectively, for n = 10, 50, 100, and b = 10. It is clear that the new operator exhibits faster convergence towards the limit than the old operator. Also, the new operator is giving flexibility in choosing parameters in the form of the sequences μ n and ν n .

Conclusions
We have modified the sequence of operators discussed in [9] and developed many approximation properties such as direct theorems, rate of convergence in weighted spaces, and approximation for functions of bounded variation. Moreover, we have also shown the convergence of old and modified new operators graphically.