Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.


Introduction
In the year 1950, a famous mathematician Szász [1] invented the positive linear operators for the continuous function f on ½0, ∞Þ and that were extensively searched rather than Bernstein operators [2]. For z ∈ ½0,∞Þ and f ∈ C½0,∞Þ, Szász introduced the operators as follows: where C½0, ∞Þ is the space of continuous functions on ½0, ∞Þ.

Approximations in Weighted Space
In the present section, we follow the well-known results by Gadziev [21] and recall the results in weighted spaces with some additional conditions precisely, under the analogous of P.P. Korovkin's theorem holds. In order to define the uniformly approximations, we take z → φðzÞ be the kind of functions which is continuous and strictly increasing with the assumptions ΦðzÞ = 1 + φ 2 ðzÞ and lim z→∞ ΦðzÞ = ∞. For this reason, we let B Φ ½0, ∞Þ be a set of all such functions which are defined on ½0, ∞Þ and verifying the results where K f is a constant and depending only on function f and B Φ ½0, ∞Þ equipped the norm with 3 Journal of Function Spaces Furthermore, we denote the set all continuous functions on ½0, ∞Þ by C½0, ∞Þ and its subsets be C Φ ½0, ∞Þ defined as C Φ ½0,∞Þ = B Φ ½0,∞Þ ∩ C½0,∞Þ.
It is well known for the sequence of linear positive operators fK r g r≥1 (see [21] where M is a positive constant. For m ∈ ℕ, let us denote where ⇒ denotes the uniform convergence.
Theorem 5 [21,23]. Let the positive linear operators fJ r g r≥1 acting from C Φ ½0, ∞Þ to B Φ ½0, ∞Þ and for Proof. It is enough to prove Theorem 6; we use the wellknown Korovkin theorem and show Taking into account Lemma 2, then it is easy to see that For j = 1, we can write here If r → ∞, then easily we get ∥R * r,η ðγ 1 Þ − z∥ Φ → 0. Similarly, for j = 2, we conclude that Thus, we easily get ∥R * r,η ðγ 2 Þ − z 2 ∥ Φ → 0, as r → ∞.
Proof. By the virtue of |φðzÞ | ≤kφk Φ ð1 + z 2 Þ and for any positive real z 0 , we easily obtain Thus, Now, for each ε > 0, there exists r 1 ∈ ℕ for all r ≥ r 1 such that Therefore, for all r ≥ r 1 In view of (26) and (29), we get If we choose any z 0 so large, such that kφk Φ /ð1 + z 2 0 Þ ξ ≤ ε/6, then we get On the other hand, there exists r 2 ≥ r such that Finally, take r 3 = max ðr 1 , r 2 Þ and on combining (31) and (32) with the above expression, we get This completes the proof of Theorem 7.
Proof. If we consider Lemma 2 and Theorem 9, then we can obtain

Journal of Function Spaces
Proof. If we consider Lemmas 2 and 3 and (2) of Theorem 9, then it is obvious to get that  [25] for an arbitrary f ∈ C m Φ ½0,∞Þ, m ∈ ℕ ∪ f0g, the weighted modulus of continuity introduced such that Two main properties of this modulus of continuity are lim δ→0 Ωð f ; δÞ = 0 and where t, z ∈ ½0, ∞Þ and Ω weighted modulus of continuity of the function for f ∈ C m Φ ½0,∞Þ.
Theorem 12. Let f ∈ C m Φ ½0,∞Þ, then for all z ∈ ½0,∞Þ we have the inequality where M = 2ð2 + M 1 + ffiffiffiffiffiffi ffi M 2 p Þ > 0, for M 1 , M 2 > 0 and Proof. We use expressions (41) and (42) and applying the Cauchy-Schwarz inequality to operators R * r,η , we get We know the expression In view of Lemma 3, we can obtain where M 1 and M 2 are positive constant and If we choose δ = ffiffiffiffiffiffiffiffiffiffi ffi α r ðηÞ p and taking supremum z ∈ ½0, α r ðηÞÞ, then we easily get the result.

Direct Approximation Results of R * r,η
The present section gives some direct approximation results in space of K-functional and in Lipschitz spaces. We take C b ½0, ∞Þ be the set of all continuous and bounded functions defined on ½0, ∞Þ.
Definition 13. For every δ > 0 and f ∈ C½0, ∞Þ the K -functional is defined such that For an absolute constant M > 0, one has Let ω 2 ð f ; δÞ denote the modulus of continuity of order two such that while the classical modulus of continuity is given by Theorem 14. For an arbitrary φ ∈ C 2 b ½0,∞Þ, let an auxiliary operator S * r,η be such that Then, for any ϕ ∈ C 2 b ½0,∞Þ operators (55), verify the inequality where δ * r,η ðzÞ is defined by Theorem 10.
Proof. For any ϕ ∈ C 2 b ½0,∞Þ, it is easy to verify that K * r,η ðγ 0 ; zÞ = 1 and We have For any ϕ ∈ C 2 b ½0,∞Þ, the Taylor series expression gives us Therefore, after applying the operators K * r,η , on both Thus, we get This gives the complete proof.
Proof. We prove Theorem 15 in view of Theorem 14. Therefore, for all f ∈ C b ½0,∞Þ and ϕ ∈ C 2 b ½0,∞Þ, we get If we take infimum for all ϕ ∈ C 2 b ½0, ∞Þ, then in view of (50) it is easy to conclude that The proof is completed here.