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Introduction
J. P. King [7] introduced a modification of the well known Bernstein polynomials which preserve constant and the x2 test function. This modification provide better approximation over the usual Bernstein polynomials. In addition, a modification of Szászirakyan operators is presented that reproduces the functions 1 and e2ax, a > 0 fixed and also are proved uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya-type theorem by T. Acar [13]. Many different applications of similar type of operators have studied in [7]-[9].
In [1], Jakimovski and Leviatan constructed a new type of operators Pn by using Appell polynomials given as below; $\begin{array}{}
\displaystyle
g(u)=\sum_{n=0}^\infty a_{n}u^n, g(1)\neq 1
\end{array}
$ be an analytic function in the disk $\begin{array}{}
\displaystyle
p_{k}(x)=\sum_{i=0}^k a_{i}\frac{x^{k-i}}{(k-i)!},
\end{array}
$ k ∈ ℕ be the Appell polynomials defined by the identity
Ifg(1) = 1 in (1), we obtain$\begin{array}{}
\displaystyle
p_{k}(x)=\frac{x^k}{k!}
\end{array}$and we get classical Szász-Mirakjan operator which is given by
B. Wood in [6] proved that the operators Pn are positive if and only if $\begin{array}{}
\displaystyle
\frac{a_{n}}{g(1)}\geq 0
\end{array}
$ for n ∈ ℕ. In [5], Ciupa studied the rate of convergence of these operators. The convergence of these operators in a weighted space of functions on a positive semi-axis and estimate the approximation by using a new type of weighted modulus of continuity introduced by A.D. Gadjiev and A. Aral in [8] were studied in [3].
Main Results
In this section, we consider the following modified form of generalization of Jakimovski-Leviatan operators
In this section, we represent the rate of uniform convergence for the En operators. In [12], the uniform convergence estimate for any sequence of positive linear operators were established by Boyanov and Veselinov. In [11], Holhoş presented the following theorem:
Theorem 1
([11]) If a sequence of linear positive operators Ln:C*[0, ∞) → C*[0, ∞) satisfy the equalities
Take $\begin{array}{}
\displaystyle
r_{n}(x)=nE_{n}(\phi_{x}^1(t);x)-\frac{x}{2}
\end{array}
$ and $\begin{array}{}
\displaystyle
q_{n}(x)=\frac{1}{2}\left(nE_{n}(\phi_{x}^2(t);x)-x\right).
\end{array}$ Thus, we get
In order to complete the proof of the theorem, we must find the last term of the inequality (19). Here, we also know that the equalities given in (12) and (13)rn(x)→ 0, qn(x)→ 0 as n → ∞ at any point x ∈ [0, ∞).
If |e–x–e–t| ≤ δ, then |h(t,x)| ≤ 2ω*(f˝;δ). In case |e–x–e–t|> δ, then we get |h(t, x)| ≤ $\begin{array}{}
\displaystyle
2\frac{(e^{-t}-e^{-x})^2}{\delta^2}\omega^*(f'';\delta).
\end{array}$ Hence
In this paper, it is studied the theoretical aspects of Jakimovski-Leviatan operators which reproduce constant and e–x functions. A theorem for determining uniform convergence order of a quantitative estimate for the modified operators are presented. We also prove a quantitative Voronovskya type theorem. For the following studies, the convergence of the operators by illustrative graphics in Maple to certain functions are investigated.