The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

and Applied Analysis 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

The aim of the this paper is to study some direct results in terms of the modulus of continuity of the second order, convergence of derivative operators to derivative functions, the weighted space, and the degree of approximation of  by G ()  ,ℎ .We also study the statistical convergence.The rate of convergence of the operators G ()  ,ℎ to a certain function is also illustrated through graphics using Matlab.
The theorem below shows that the derivative (  /  )G ()  ,ℎ (; ) is also an approximation process for   /  .
where (  /  , ⋅) is the modulus of continuity of   /  .
Proof.By simple calculations, the following formula is obtained: where Δ    / ((/)  ) is the difference of order  of  corresponding to the increment   /.Using the relation between finite difference and divided difference, the derivative of order  of the operators is represented as follows: where where  3 ∈ (0, 1).Using the estimates in (15), we have the desired result.

Weighted Approximation
Let   [0, ∞) be the space of all functions  defined on [0, ∞) Since lim →∞ (  /) = 0, there exists constant   such that This proof is complete.
where () = max{ 1 ,  Proof.By simple calculation, we have Finally, from (33), we obtain If we apply Theorem 10, we obtain the desired result.

𝐴-Statistical Convergence
Now, let  = [  ], ,  ∈ N, be an infinite summability matrix.For a given sequence (  ), the -transform of , denoted by (()  ), is given by which provides the converging series for each  ∈ N. We say that  is a regular if lim  ()  =  whenever lim  (  ) = .
A sequence (  ) is called -statistically convergent to  if, for every  > 0, lim  ∑ :|  −|≥   = 0.This limit is denoted by Abstract and Applied Analysis   − lim    = .Replacing  by  1 , the Cesàro matrix of order one in ( 6), from -statistical convergence, is reduced to statistical convergence.Similarly, if we take  = , the identity matrix, then -statistical convergence coincides with the ordinary convergence.Kolk [13] proved that, in the case of lim  max  = 0, -statistical convergence is stronger than ordinary convergence.Further, we will first obtain the following weighted Korovkin theorem via -statistical convergence.
where  ∈ (, ).Thus, we get In view of (49) for  > 0, we have From (61), one can write the following: (63) Hence, taking the limit as  → ∞, we get the desired result.
The following theorem contains quantitative estimates by means of Peetre's -functional.