In this article, we purpose to study some approximation properties of the one and two variables o... more In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s - functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-ty...
In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties ... more In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties of (p,q)-Bernstein operators areinvestigated. Some basic theorems are proved. The rate of approximationby modulus of continuity is estimated.
In the present article, a modification of Jakimovski-Leviatan operators is presented which reprod... more In the present article, a modification of Jakimovski-Leviatan operators is presented which reproduce constant and e–x functions. We prove uniform convergence order of a quantitative estimate for the modified operators. We also give a quantitative Voronovskya type theorem.
We establish some approximation properties in weighted spaces and give a Voronovskaya-type asympt... more We establish some approximation properties in weighted spaces and give a Voronovskaya-type asymptotic formula for the composition of the Szăsz-Mirakyan and Durrmeyer-Chlodowsky operators.
The theorems on weighted approximation and order of approximation of continuous functions of two ... more The theorems on weighted approximation and order of approximation of continuous functions of two variables by new type Gamma operators on all positive square region are established.
In this paper we deal with the operators $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { ... more In this paper we deal with the operators $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$. Also, we establish the order of approximation by using weighted modulus of continuity.
In this article, we purpose to study some approximation properties of the one and two variables o... more In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s - functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-ty...
In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties ... more In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties of (p,q)-Bernstein operators areinvestigated. Some basic theorems are proved. The rate of approximationby modulus of continuity is estimated.
In the present article, a modification of Jakimovski-Leviatan operators is presented which reprod... more In the present article, a modification of Jakimovski-Leviatan operators is presented which reproduce constant and e–x functions. We prove uniform convergence order of a quantitative estimate for the modified operators. We also give a quantitative Voronovskya type theorem.
We establish some approximation properties in weighted spaces and give a Voronovskaya-type asympt... more We establish some approximation properties in weighted spaces and give a Voronovskaya-type asymptotic formula for the composition of the Szăsz-Mirakyan and Durrmeyer-Chlodowsky operators.
The theorems on weighted approximation and order of approximation of continuous functions of two ... more The theorems on weighted approximation and order of approximation of continuous functions of two variables by new type Gamma operators on all positive square region are established.
In this paper we deal with the operators $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { ... more In this paper we deal with the operators $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$. Also, we establish the order of approximation by using weighted modulus of continuity.
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