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Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

2022-01-01
Mohd Raiz , Amit Kumar , Vishnu Narayan Mishra +1 more
<p style='text-indent:20px;'>The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a} $\end{document}</tex-math></inline-formula>sz-Schurer-Beta type operators to approximate a class of Lebesgue 
 <p style='text-indent:20px;'>The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a} $\end{document}</tex-math></inline-formula>sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.</p>

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

2022-01-01
Lakshmi Narayan Mishra , Mohd Raiz , Laxmi Rathour +1 more
We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows 
 We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.
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Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

2023-11-11
Nadeem Rao , Mohd Raiz , M. Mursaleen +1 more
Abstract In this article, we introduce generalized beta extension of Sz $$\acute{a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>a</mml:mi><mml:mo>®</mml:mo></mml:mover></mml:math> sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these 
 Abstract In this article, we introduce generalized beta extension of Sz $$\acute{a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>a</mml:mi><mml:mo>®</mml:mo></mml:mover></mml:math> sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.
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Dunkl analouge of Sz$ \acute{a} $sz Schurer Beta bivariate operators

2022-10-10
Vishnu Narayan Mishra , Mohd Raiz , Nadeem Rao
<p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation 
 <p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation properties. Further, preliminaries results and definitions are presented. Moreover, we study existence of convergence with the aid of Korovkin theorem and order of approximation via usual modulus of continuity, Peetre's K-functional, Lipschitz maximal functional. Lastly, approximation properties of these sequences of operators are studied in B<inline-formula><tex-math id="M3">\begin{document}$ \ddot{o} $\end{document}</tex-math></inline-formula>gel space via mixed modulus of continuity.

Approximation on bivariate of Durrmeyer operators based on beta function

2023-08-25
Mohd Raiz , Ruchi Singh Rajawat , Lakshmi Narayan Mishra +1 more

SzĂĄsz-Type Operators Involving q-Appell Polynomials

2022-01-01
Mohd Raiz , Nadeem Rao , Vishnu Narayan Mishra

α-Schurer Durrmeyer operators and their approximation properties

2023-06-30
Mohd Raiz , Ruchi Singh Rajawat , Vishnu Narayan Mishra
The key goal of the present research article is to introduce a new sequence of linear positive operator i.e., α-Schurer Durrmeyer operator and their approximation behaviour on the basis of 
 The key goal of the present research article is to introduce a new sequence of linear positive operator i.e., α-Schurer Durrmeyer operator and their approximation behaviour on the basis of function η (z), where η infinitely differentiable on [0,1], η(z)=0, η(1)=1 and η'(z)&gt;0, for all z∈[0,1]. Further, we calculate central moments and basic estimates for the sequence of the operators. Moreover, we discuss the rate of convergence and order of approximation in terms of modulus of continuity, smoothness, Korovkin theorem, and Peeter's K-functional. Lastly, local and global approximation properties are studied in the subsequent section.
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Symmetric Properties of λ-Szåsz Operators Coupled with Generalized Beta Functions and Approximation Theory

2024-12-22
Nadeem Rao , Mohammad Farid , Mohd Raiz
This research work focuses on λ-SzĂĄsz–Mirakjan operators coupling generalized beta function. The kernel functions used in λ-SzĂĄsz operators often possess even or odd symmetry. This symmetry influences the behavior of 
 This research work focuses on λ-SzĂĄsz–Mirakjan operators coupling generalized beta function. The kernel functions used in λ-SzĂĄsz operators often possess even or odd symmetry. This symmetry influences the behavior of the operator in terms of approximation and convergence properties. The convergence properties, such as uniform convergence and pointwise convergence, are studied in view of the Korovkin theorem, the modulus of continuity, and Peetre’s K-functional of these sequences of positive linear operators in depth. Further, we extend our research work for the bivariate case of these sequences of operators. Their uniform rate of approximation and order of approximation are investigated in Lebesgue measurable spaces of function. The graphical representation and numerical error analysis in terms of the convergence behavior of these operators are studied.

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

2021-01-01
Lakshmi Narayan Mishra , Mohd Raiz , Vishnu Narayan Mishra
We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows 
 We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.
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Approximation behaviour of generalized Baskakov-Durrmeyer-Schurer operators

2024-06-30
Nadeem Rao , Mohd Raiz , Vishnu Narayan Mishra
The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of 
 The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of approximation are studied in terms of first and second-order modulus of continuity. We prove a Korovkin-type approximation theorem and obtain the rate of convergence of these operators. Moreover, local and global approximation properties are discussed in different functional spaces. Lastly, A-statistical approximation results are presented.

Quantitative means of an asymptotic formula for a family of modified linear positive operators

2024-01-01
Rishikesh Yadav , Mohd Raiz , Vishnu Narayan Mishra

Convergence properties of the limit q-Baskakov operators

2024-01-01
Asha Ram Gairola , Amrita Singh , Mohd Raiz +1 more

Estimates of bivariate new Kantorovich type of the BalĂĄzs-Szabados operators based on q-integers

2025-01-01
Hayatem Hamal , Mohd Raiz

Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

2025-03-27
Nadeem Rao , Mohammad Farid , Mohd Raiz
This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and 
 This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence.

On the Approximations and Symmetric Properties of Frobenius–Euler–ƞimƟek Polynomials Connecting Szász Operators

2025-04-25
Nadeem Rao , Mohammad Farid , Mohd Raiz
This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. 
 This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions.

Approximation by Bicomplex Favard–Szász–Mirakjan Operators

2025-05-30
George A. Anastassiou , Özge Özalp GĂŒller , Mohd Raiz +1 more
The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact C2 disk. We provide quantitative estimates of the convergence. Moreover, the 
 The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact C2 disk. We provide quantitative estimates of the convergence. Moreover, the Voronovskaja-type results for analytic functions and the simultaneous approximation by bicomplex Favard–Szász–Mirakjan operators are investigated.

Approximation by Bicomplex Favard–Szász–Mirakjan Operators

2025-05-30
George A. Anastassiou , Özge Özalp GĂŒller , Mohd Raiz +1 more
The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact C2 disk. We provide quantitative estimates of the convergence. Moreover, the 
 The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact C2 disk. We provide quantitative estimates of the convergence. Moreover, the Voronovskaja-type results for analytic functions and the simultaneous approximation by bicomplex Favard–Szász–Mirakjan operators are investigated.

On the Approximations and Symmetric Properties of Frobenius–Euler–ƞimƟek Polynomials Connecting Szász Operators

2025-04-25
Nadeem Rao , Mohammad Farid , Mohd Raiz
This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. 
 This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions.

Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

2025-03-27
Nadeem Rao , Mohammad Farid , Mohd Raiz
This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and 
 This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence.

Estimates of bivariate new Kantorovich type of the BalĂĄzs-Szabados operators based on q-integers

2025-01-01
Hayatem Hamal , Mohd Raiz

Symmetric Properties of λ-Szåsz Operators Coupled with Generalized Beta Functions and Approximation Theory

2024-12-22
Nadeem Rao , Mohammad Farid , Mohd Raiz
This research work focuses on λ-SzĂĄsz–Mirakjan operators coupling generalized beta function. The kernel functions used in λ-SzĂĄsz operators often possess even or odd symmetry. This symmetry influences the behavior of 
 This research work focuses on λ-SzĂĄsz–Mirakjan operators coupling generalized beta function. The kernel functions used in λ-SzĂĄsz operators often possess even or odd symmetry. This symmetry influences the behavior of the operator in terms of approximation and convergence properties. The convergence properties, such as uniform convergence and pointwise convergence, are studied in view of the Korovkin theorem, the modulus of continuity, and Peetre’s K-functional of these sequences of positive linear operators in depth. Further, we extend our research work for the bivariate case of these sequences of operators. Their uniform rate of approximation and order of approximation are investigated in Lebesgue measurable spaces of function. The graphical representation and numerical error analysis in terms of the convergence behavior of these operators are studied.

Approximation behaviour of generalized Baskakov-Durrmeyer-Schurer operators

2024-06-30
Nadeem Rao , Mohd Raiz , Vishnu Narayan Mishra
The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of 
 The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of approximation are studied in terms of first and second-order modulus of continuity. We prove a Korovkin-type approximation theorem and obtain the rate of convergence of these operators. Moreover, local and global approximation properties are discussed in different functional spaces. Lastly, A-statistical approximation results are presented.

Quantitative means of an asymptotic formula for a family of modified linear positive operators

2024-01-01
Rishikesh Yadav , Mohd Raiz , Vishnu Narayan Mishra

Convergence properties of the limit q-Baskakov operators

2024-01-01
Asha Ram Gairola , Amrita Singh , Mohd Raiz +1 more

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

2023-11-11
Nadeem Rao , Mohd Raiz , M. Mursaleen +1 more
Abstract In this article, we introduce generalized beta extension of Sz $$\acute{a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>a</mml:mi><mml:mo>®</mml:mo></mml:mover></mml:math> sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these 
 Abstract In this article, we introduce generalized beta extension of Sz $$\acute{a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>a</mml:mi><mml:mo>®</mml:mo></mml:mover></mml:math> sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.
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Approximation on bivariate of Durrmeyer operators based on beta function

2023-08-25
Mohd Raiz , Ruchi Singh Rajawat , Lakshmi Narayan Mishra +1 more

α-Schurer Durrmeyer operators and their approximation properties

2023-06-30
Mohd Raiz , Ruchi Singh Rajawat , Vishnu Narayan Mishra
The key goal of the present research article is to introduce a new sequence of linear positive operator i.e., α-Schurer Durrmeyer operator and their approximation behaviour on the basis of 
 The key goal of the present research article is to introduce a new sequence of linear positive operator i.e., α-Schurer Durrmeyer operator and their approximation behaviour on the basis of function η (z), where η infinitely differentiable on [0,1], η(z)=0, η(1)=1 and η'(z)&gt;0, for all z∈[0,1]. Further, we calculate central moments and basic estimates for the sequence of the operators. Moreover, we discuss the rate of convergence and order of approximation in terms of modulus of continuity, smoothness, Korovkin theorem, and Peeter's K-functional. Lastly, local and global approximation properties are studied in the subsequent section.
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Dunkl analouge of Sz$ \acute{a} $sz Schurer Beta bivariate operators

2022-10-10
Vishnu Narayan Mishra , Mohd Raiz , Nadeem Rao
<p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation 
 <p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation properties. Further, preliminaries results and definitions are presented. Moreover, we study existence of convergence with the aid of Korovkin theorem and order of approximation via usual modulus of continuity, Peetre's K-functional, Lipschitz maximal functional. Lastly, approximation properties of these sequences of operators are studied in B<inline-formula><tex-math id="M3">\begin{document}$ \ddot{o} $\end{document}</tex-math></inline-formula>gel space via mixed modulus of continuity.

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

2022-01-01
Lakshmi Narayan Mishra , Mohd Raiz , Laxmi Rathour +1 more
We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows 
 We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.
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Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

2022-01-01
Mohd Raiz , Amit Kumar , Vishnu Narayan Mishra +1 more
&lt;p style='text-indent:20px;'&gt;The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \acute{a} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;sz-Schurer-Beta type operators to approximate a class of Lebesgue 
 &lt;p style='text-indent:20px;'&gt;The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \acute{a} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.&lt;/p&gt;

SzĂĄsz-Type Operators Involving q-Appell Polynomials

2022-01-01
Mohd Raiz , Nadeem Rao , Vishnu Narayan Mishra

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

2021-01-01
Lakshmi Narayan Mishra , Mohd Raiz , Vishnu Narayan Mishra
We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows 
 We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces($IFNS$), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in $IFNS$ follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.
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Coauthor Papers Together
Vishnu Narayan Mishra 11
Nadeem Rao 8
Lakshmi Narayan Mishra 3
Mohammad Farid 3
Ruchi Singh Rajawat 2
M. Mursaleen 1
Laxmi Rathour 1
Seda Karateke 1
Asha Ram Gairola 1
Amrita Singh 1
Özge Özalp GĂŒller 1
Amit Kumar 1
Rishikesh Yadav 1
Hayatem Hamal 1
George A. Anastassiou 1

Dunkl analouge of Sz$ \acute{a} $sz Schurer Beta bivariate operators

2022-10-10
Vishnu Narayan Mishra , Mohd Raiz , Nadeem Rao
<p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation 
 <p style='text-indent:20px;'>The motive of this research article is to introduce a sequence of Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a}sz $\end{document}</tex-math></inline-formula> Schurer Beta bivariate operators in terms of generalization exponential functions and their approximation properties. Further, preliminaries results and definitions are presented. Moreover, we study existence of convergence with the aid of Korovkin theorem and order of approximation via usual modulus of continuity, Peetre's K-functional, Lipschitz maximal functional. Lastly, approximation properties of these sequences of operators are studied in B<inline-formula><tex-math id="M3">\begin{document}$ \ddot{o} $\end{document}</tex-math></inline-formula>gel space via mixed modulus of continuity.

Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

2022-01-01
Mohd Raiz , Amit Kumar , Vishnu Narayan Mishra +1 more
&lt;p style='text-indent:20px;'&gt;The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \acute{a} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;sz-Schurer-Beta type operators to approximate a class of Lebesgue 
 &lt;p style='text-indent:20px;'&gt;The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \acute{a} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.&lt;/p&gt;

Local and global results for modified Szász–Mirakjan operators

2016-09-27
Rajiv Gandhi , Vishnu Narayan Mishra
In this paper, we study a natural modification of Sz\'{a}sz - Mirakjan operators. It is shown by discussing many important established results for Sz\'{a}sz - Mirakjan operators. The results do 
 In this paper, we study a natural modification of Sz\'{a}sz - Mirakjan operators. It is shown by discussing many important established results for Sz\'{a}sz - Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations.
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Construction of a new family of Bernstein‐Kantorovich operators

2017-08-17
S. A. Mohiuddine , Tuncer Acar , Abdullah Alotaibi
In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α . The uniform convergence of the operators and rate of convergence in local 
 In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α . The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors.

Generalization of Bernstein's polynomials to the infinite interval

1950-09-01
Otto SzĂĄsz
The paper studi es t he convergence of P (u, x) to f (x) as u -> 00 .The resul ts obtained are generalized anal ogs, for the interval 0 
 The paper studi es t he convergence of P (u, x) to f (x) as u -> 00 .The resul ts obtained are generalized anal ogs, for the interval 0 :

On Lipschitz-type maximal functions and their smoothness spaces

1988-01-01
Burkhard Lenze

Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter $${\varvec{\alpha }}$$

2020-01-16
S. A. Mohiuddine , Faruk Özger

Approximation properties of bivariate SzĂĄsz Durrmeyer operators via Dunkl analogue

2021-01-01
Nadeem Rao , Md Heshamuddin , Mohd Shadab +1 more
In the present article, we construct a new sequence of bivariate Sz?sz-Durrmeyer operators based on Dunkl analogue. We investigate the order of approximation with the aid of modulus of continuity 
 In the present article, we construct a new sequence of bivariate Sz?sz-Durrmeyer operators based on Dunkl analogue. We investigate the order of approximation with the aid of modulus of continuity in terms of well known Peetre?s K-functional, weighted approximation results, Voronovskaja type theorems and Lipschitz maximal functions. Further, we also discuss here the approximation properties of the operators in B?gel-spaces by use of mixed-modulus of continuity.
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Improved rate of approximation by modification of Baskakov operator

2022-01-01
Animesh Gairola , Amrita Singh , Laxmi Rathour +1 more
The optimal order of approximation, |L n f (x)f (x)| of a linear positive operator L n f (x) is 1/n and can not be improved however smooth the function 
 The optimal order of approximation, |L n f (x)f (x)| of a linear positive operator L n f (x) is 1/n and can not be improved however smooth the function may be.We remove the positivity of the Baskakov operator V n ( f ;x) and introduce its three variants V M,i n ( f ;x) , i = 1,2,3.We prove that the rates of approximation by these operators are improved from the linear order 1/n to quadratic order 1/n 2 and then to cubic order 1/n 3 for sufficiently smooth functions.
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Blending‐type approximation by LupaƟ–Durrmeyer‐type operators involving Pólya distribution

2021-04-08
Arun Kajla , S. A. Mohiuddine , Abdullah Alotaibi
In the present work, we construct a new sequence of positive linear operatorsinvolving PĂłlya distribution. We compute a Voronovskaja type and a GrĂŒss–Voronovskaja type asymptotic formula as well as the 
 In the present work, we construct a new sequence of positive linear operatorsinvolving PĂłlya distribution. We compute a Voronovskaja type and a GrĂŒss–Voronovskaja type asymptotic formula as well as the rate of approximation by using the modulus of smoothness and for functions in a Lipschitz type space. Lastly, we provide some numerical results, which explain the validity of the theoretical results and the effectiveness of the constructed operators.

Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators

2013-12-01
Vishnu Narayan Mishra , Kejal Khatri , Lakshmi Narayan Mishra +1 more
In the present paper, we study an inverse result in simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators. MSC:41A25, 41A35, 41A36. In the present paper, we study an inverse result in simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators. MSC:41A25, 41A35, 41A36.
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Local approximation properties for certain King type operators

2013-01-01
Ali Özarslan , HĂŒseyin Aktuğlu
In this paper, we consider a certain King type operators which includes general families of Szász-Mirakjan, Baskakov, Post-Widder and Stancu operators. By introducing two parameter family of Lipschitz type space, 
 In this paper, we consider a certain King type operators which includes general families of Szász-Mirakjan, Baskakov, Post-Widder and Stancu operators. By introducing two parameter family of Lipschitz type space, which provides global approximation for the above mentioned operators, we obtain the rate of convergence of this class. Furthermore, we give local approximation results by using the first and the second modulus of continuity.
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Korovkin-type Approximation Theory and its Applications

1994-12-31
Francesco Altomare , Michele Campıtı

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

2020-12-01
S. A. Mohiuddine
Abstract We construct the bivariate form of Bernstein–Schurer operators based on parameter α . We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the 
 Abstract We construct the bivariate form of Bernstein–Schurer operators based on parameter α . We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K -functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.
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SzĂĄsz-Type Operators Involving q-Appell Polynomials

2022-01-01
Mohd Raiz , Nadeem Rao , Vishnu Narayan Mishra

Better Uniform Approximation by New Bivariate Bernstein Operators

2022-11-07
Asha Ram Gairola , Suruchi Maindola , Laxmi Rathour +2 more
In this paper we introduce new bivariate Bernstein type operators BnM,i(f; x, y), i = 1, 2, 3. The rates of approximation by these operators are calculated and it is 
 In this paper we introduce new bivariate Bernstein type operators BnM,i(f; x, y), i = 1, 2, 3. The rates of approximation by these operators are calculated and it is shown that the errors are significantly smaller than those of ordinary bivariate Bernstein operators for sufficiently smooth functions.
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Dunkl analogue of Szasz operators

2014-07-17
Sezgi̇n Sucu

Approximation on bivariate parametric-extension of Baskakov-Durrmeyer-operators

2021-01-01
Md. Nasiruzzaman , Nadeem Rao , Manish Kumar +1 more
The main purpose of this article is to study the bivariate approximation generalization for Baskakov-Durrmeyer-operators with the aid of non-negative parametric variants suppose 0 ? ?1,?2 ? 1. We obtain 
 The main purpose of this article is to study the bivariate approximation generalization for Baskakov-Durrmeyer-operators with the aid of non-negative parametric variants suppose 0 ? ?1,?2 ? 1. We obtain the order of approximation by use of the modulus of continuity in terms of well known Peetre?s K-functional, Voronovskaja type theorems and Lipschitz maximal functions. Further, we also discuss here the approximation properties of the operators in B?gel-spaces by use of mixed-modulus of continuity.
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Some approximation properties of bivariate Bleimann–Butzer–Hahn operators based on (p, q)-integers

2016-05-18
M. ‎Mursaleen , Md. Nasiruzzaman

A Dunkl type generalization of SzĂĄsz operators via post-quantum calculus

2018-10-22
Abdullah Alotaibi , Md. Nasiruzzaman , M. ‎Mursaleen
The object of this paper to construct Dunkl type Szász operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by 
 The object of this paper to construct Dunkl type Szász operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. We also study the bivariate version of these operators.
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Approximation by the Parametric Generalization of Baskakov–Kantorovich Operators Linking with Stancu Operators

2021-01-05
S. A. Mohiuddine , Naeem Ahmad , Faruk Özger +2 more

Some Approximation Theorems via Statistical Convergence

2002-03-01
Akif D. Gadjiev , Cihan Orhan
In this paper we prove some Korovkin and Weierstrass type approximation theorems via statistical convergence.We are also concerned with the order of statistical convergence of a sequence of positive linear 
 In this paper we prove some Korovkin and Weierstrass type approximation theorems via statistical convergence.We are also concerned with the order of statistical convergence of a sequence of positive linear operators.
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Order of Approximation by a New Univariate Kantorovich Type Operator

2023-09-25
Asha Ram Gairola , Nidhi Bisht , Laxmi Rathour +2 more
In order to approximate Lebesgue integrable functions on [0, 1], a sequence of linear positive integral operators of Kantorovich type Lσ&lt;sσ&gt;f (x) with a parameter sσ is introduced. The estimates 
 In order to approximate Lebesgue integrable functions on [0, 1], a sequence of linear positive integral operators of Kantorovich type Lσ&lt;sσ&gt;f (x) with a parameter sσ is introduced. The estimates for rates of approximation for functions with a specific smoothness are proved using the appropriate modulus of continuity.
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SzĂĄsz-Durrmeyer Operators Based on Dunkl Analogue

2017-02-23
Abdul Wafi , Nadeem Rao

Generalized Hermite Polynomials and the Bose-Like Oscillator Calculus

1994-01-01
Marvin Rosenblum
This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the 
 This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator calculus, which generalizes the calculus associated with the quantum mechanical simple harmonic oscillator, is studied in terms of these polynomials.
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Statistical approximation by Kantorovich-type discrete q-Betaoperators

2013-11-29
Vishnu Narayan Mishra , Kejal Khatri , Lakshmi Narayan Mishra
The aim of the present paper is to introduce a Kantorovich-type modification ofthe q-discrete beta operators and to investigate their statistical andweighted statistical approximation properties. Rates of statistical convergenceby means 
 The aim of the present paper is to introduce a Kantorovich-type modification ofthe q-discrete beta operators and to investigate their statistical andweighted statistical approximation properties. Rates of statistical convergenceby means of the modulus of continuity and the Lipschitz-type function are alsoestablished for operators. Finally, we construct a bivariate generalization ofthe operator and also obtain the statistical approximation properties. MSC: 41A25, 41A36.
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On the order of approximation by modified summation-integral-type operators based on two parameters

2023-01-01
S. A. Mohiuddine , Karunesh Kumar Singh , Abdullah Alotaibi
Abstract In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, 
 Abstract In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-SzĂĄsz, SzĂĄsz-Beta, LupaƟ-Beta, LupaƟ-SzĂĄsz, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise GrĂŒss-Voronovskaja-type result and also establish the GrĂŒss-Voronovskaja-type asymptotic formula in quantitative form.
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Approximation Process Based on Parametric Generalization of Schurer–Kantorovich Operators and their Bivariate Form

2022-05-09
Md. Nasiruzzaman , H. M. Srivastava , S. A. Mohiuddine

Bernstein Polynomials

1993-01-01
Ronald DeVore , George G. Lorentz

SzĂĄsz type operators involving Charlier polynomials and approximation properties

2021-01-01
Ahmed Ahmed Hussin Ali Al-Abied , M. Mursaleen , M. ‎Mursaleen
Our aim is to define modified Sz?sz type operators involving Charlier polynomials and obtain some approximation properties. We prove some results on the order of convergence by using the modulus 
 Our aim is to define modified Sz?sz type operators involving Charlier polynomials and obtain some approximation properties. We prove some results on the order of convergence by using the modulus of smoothness and Peetre?s K-functional. We also establish Voronoskaja type theorem for these operators. Moreover, we prove a Korovkin type approximation theorem via q-statistical convergence.
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Approximation by bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS operators on weighted spaces

2022-02-23
ReƟat Aslan , M. ‎Mursaleen
Abstract In this article, we consider a bivariate Chlodowsky type SzĂĄsz–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of 
 Abstract In this article, we consider a bivariate Chlodowsky type SzĂĄsz–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of continuity and also for the elements of the Lipschitz type class. Moreover, we examine the degree of convergence with regard to the weighted modulus of continuity and Peetre’s K -functional. Further, we construct the associated GBS type of these operators and estimate the degree of approximation using the mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, with the help of Maple software, we present the comparisons of the convergence of the bivariate Chlodowsky type SzĂĄsz–Durrmeyer operators and associated GBS type operators to certain functions with some graphs and error estimation tables.
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Numerical and theoretical approximation results for Schurer–Stancu operators with shape parameter $$ \lambda $$

2022-05-17
Khursheed J‎. ‎Ansari , Faruk Özger , Zeynep ÖdemiƟ Özger

Approximation by modified Bernstein polynomials based on real parameters

2023-02-15
Ruchi Singh Rajawat , Karunesh Kumar Singh , Vishnu Narayan Mishra
In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, 
 In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators.
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Inverse theorems for some linear positive operators using Beta and Baskakov basis functions

2021-01-01
Lakshmi Narayan Mishra , Anshul Srivastava , Taqseer Khan +2 more
In this paper order of approximation has been improved by taking linear combination of Stancu type generalization of hybrid linear positive operators by combining Beta and Baskakov basis functions. In 
 In this paper order of approximation has been improved by taking linear combination of Stancu type generalization of hybrid linear positive operators by combining Beta and Baskakov basis functions. In this paper, we will disscuss some inverse results for our operators with help of Fubini's theorem.
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Approximation Properties of Bivariate Extension of q-Bernstein–Schurer–Kantorovich operators

2015-02-12
Ana Maria Acu , Carmen Violeta Muraru

Approximation of functions by a new class of generalized Bernstein–Schurer operators

2020-07-21
Faruk Özger , H. M. Srivastava , S. A. Mohiuddine

Estimates for the differences of positive linear operators and their derivatives

2019-10-28
Ana Maria Acu , Ioan RaƟa
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Applications of q-Calculus in Operator Theory

2013-01-01
Ali̇ Aral , Vijay Gupta , Ravi P. Agarwal

q-Szász–Durrmeyer Type Operators Based on Dunkl Analogue

2018-06-28
Nadeem Rao , Abdul Wafi , Ana Maria Acu

A generalized Dunkl type modifications of Phillips operators

2018-11-22
Md. Nasiruzzaman , Nadeem Rao
The main purpose of this present article is to discuss the convergence of Lebesgue measurable functions by providing a Dunkl generalization of Szász type operators known as Phillips operators. To 
 The main purpose of this present article is to discuss the convergence of Lebesgue measurable functions by providing a Dunkl generalization of Szász type operators known as Phillips operators. To achieve the results of a better way of uniform convergence of the Phillips operators, we study qualitative results in a Korovkin and weighted Korovkin space.
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Statistical approximation by positive linear operators

2004-01-01
Oktay Duman , Cihan Orhan
Using $A$-statistical convergence, we prove a Korovkin type approximation theorem which concerns the problem of approximating a function $f$ by means of a sequence $\{T_{n}(f;x)\}$ of positive linear operators acting 
 Using $A$-statistical convergence, we prove a Korovkin type approximation theorem which concerns the problem of approximating a function $f$ by means of a sequence $\{T_{n}(f;x)\}$ of positive linear operators acting from a weighted space $C_{\varrho_{1}}
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Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators

2019-01-01
Faruk Özger
In this study, we consider statistical approximation properties of univariate and bivariate ?-Kantorovich operators. We estimate rate of weighted A-statistical convergence and prove a Voronovskajatype approximation theorem by a family 
 In this study, we consider statistical approximation properties of univariate and bivariate ?-Kantorovich operators. We estimate rate of weighted A-statistical convergence and prove a Voronovskajatype approximation theorem by a family of linear operators using the notion of weighted A-statistical convergence. We give some estimates for differences of ?-Bernstein and ?-Durrmeyer, and ?-Bernstein and ?-Kantorovich operators. We establish a Voronovskaja-type approximation theorem by weighted A-statistical convergence for the bivariate case.
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Approximation and Geometric Properties of Complex $$\alpha $$ α -Bernstein Operator

2019-01-31
Nursel Çetіn

A generalization of SzĂĄsz-type operators which preserves constant and quadratic test functions

2016-09-06
Abdul Wafi , Nadeem Rao
In the present article, we introduced a new form of Szász-type operators which preserves test functions e0 and e2(ei(t)=ti,i=0,2). By these sequence of positive linear operators, we gave rate of 
 In the present article, we introduced a new form of Szász-type operators which preserves test functions e0 and e2(ei(t)=ti,i=0,2). By these sequence of positive linear operators, we gave rate of convergence and better error estimation by means of modulus of continuity. Moreover, we have discussed order of approximation with the help of local results. In the last, weighted Korovkin theorem is established.

Approximation by Kantorovich Type q-Bernstein-Stancu Operators

2016-06-15
M. ‎Mursaleen , Khursheed J‎. ‎Ansari , Asif Khan
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Thresholded spectral algorithms for sparse approximations

2017-01-23
Zheng-Chu Guo , Dao-Hong Xiang , Xin Guo +1 more
Spectral algorithms form a general framework that unifies many regularization schemes in learning theory. In this paper, we propose and analyze a class of thresholded spectral algorithms that are designed 
 Spectral algorithms form a general framework that unifies many regularization schemes in learning theory. In this paper, we propose and analyze a class of thresholded spectral algorithms that are designed based on empirical features. Soft thresholding is adopted to achieve sparse approximations. Our analysis shows that without sparsity assumption of the regression function, the output functions of thresholded spectral algorithms are represented by empirical features with satisfactory sparsity, and the convergence rates are comparable to those of the classical spectral algorithms in the literature.
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Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

2015-10-03
P. Ν. Agrawal , Nurhayat İspir

Bivariate generalization of <i>q</i>-Bernstein-Kantorovich type operator

2016-03-25
Preeti Sharma , Vishnu Narayan Mishra
In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman–Korovkin-type approximation theorem of these operators. We also compute the rate of 
 In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman–Korovkin-type approximation theorem of these operators. We also compute the rate of convergence using the first modulus of smoothness.

GrĂŒss-type and Ostrowski-type inequalities in approximation theory

2011-11-01
Ana Maria Acu , Heiner Gonska , Ioan RaƟa
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A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

2019-02-15
Md. Nasiruzzaman , Aiman Mukheimer , M. ‎Mursaleen
In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also 
 In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators.
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