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Öğe Applications of (p,q)-gamma function to Szász Durrmeyer operators(Publications L Institut Mathematique Matematicki, 2017) Aral, Ali; Gupta, VijayWe define a (p, q) analogue of Gamma function. As an application, we propose (p, q)-Szasz-Durrmeyer operators, estimate moments and establish some direct results.Öğe Approximation by q Baskakov Beta operators(Springer Heidelberg, 2011) Gupta, Vijay; Aral, AliIn the present paper we introduce the q analogue of the Baskakov Beta operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0, a). Then we obtain point-wise estimate, using the Lipschitz type maximal function.Öğe Approximation by Some Baskakov-Kantorovich Exponential-Type Operators(Springer Singapore Pte Ltd, 2022) Özsaraç, Fırat; Gupta, Vijay; Aral, AliIn the present paper, we propose the modification of the Baskakov-Kantorovich operators based on mu-integral. Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity.Öğe Bernstein durrmeyer operators based on two parameters(Univ Nis, 2016) Gupta, Vijay; Aral, AliIn the present paper, we study the applications of the extension of quantum calculus based on two parameters. We define beta function and establish an identity with gamma function, for two parameters (p, q), ie. the post-quantum calculus. We also propose the (p, q)-Durrmeyer operators, estimate moments and establish some direct results. Depending on the selection of p and q, the rate of convergence of the our new operators can provide better approximation than those of the Bernstein-Durrmeyer operators and its q-analogue. In the end, we provide some graphs using the software Mathematica.Öğe Convergence of the q analogue of Szasz-Beta operators(Elsevier Science Inc, 2010) Gupta, Vijay; Aral, AliIn the present paper we introduce the q analogue of the well known Szasz-Beta operators [11]. We also establish the approximation properties of these operators and estimate convergence results. In the end we propose an open problem. (C) 2010 Elsevier Inc. All rights reserved.Öğe Direct Estimates for Lupas-Durrmeyer Operators(Univ Nis, Fac Sci Math, 2016) Aral, Ali; Gupta, VijayThe generalization of the Bernstein polynomials based on Polya distribution was first considered by Stancu [14]. Very recently Gupta and Rassias [6] proposed the Durrmeyer type modification of the Lupas, operators and established some results. Now we extend the studies and here we estimate the convergence estimates, which include quantitative asymptotic formula and rate of approximation bounded variation. We also give an open problem for readers to obtain the moments using hypergeometric function.Öğe Generalized Baskakov-Beta Operators(Rocky Mt Math Consortium, 2009) Gupta, Vijay; Aral, AliVery recently Wang [9] introduced the modified form of Baskakov-beta operators and obtained a Voronov-skaja type asymptotic formula for these operators. We extend the study and here we estimate a direct result in terms of higher order modulus of continuity and an inverse theorem in simultaneous approximation for these new modified Baskakov-beta operators.Öğe Generalized q-Baskakov operators(Walter De Gruyter Gmbh, 2011) Aral, Ali; Gupta, VijayIn the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.Öğe Modification of Exponential Type Operators Preserving Exponential Functions Connected with x3(Springer Basel Ag, 2021) Gupta, Vijay; Aral, Ali; Muraru, Carmen-Violetan the present paper, we propose modification of the exponential type operators, which are connected with x(3). Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity.Öğe Note on Szasz-Mirakyan-Durrmeyer Operators Preserving e(2ax), a > 0(Taylor & Francis Inc, 2018) Deniz, Emre; Aral, Ali; Gupta, VijayIn the current article, we study Szasz-Mirakyan-Durrmeyer operators which reproduces constant and e(2ax), a > 0 functions. We discuss a uniform estimate and estimate a quantitative asymptotic formula for the modified operators.Öğe A note on Szasz-Mirakyan-Kantorovich type operators preserving e(-x)(Springer, 2018) Gupta, Vijay; Aral, AliIn the present article, we study modified form of Szasz-Mirakyan-Kantorovich operators, which reproduce constant and functions. We discuss a uniform convergence result along with a quantitative estimate for the modified operators.Öğe On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators(Walter De Gruyter Gmbh, 2015) Acar, Tuncer; Aral, Ali; Gupta, VijayThe present paper deals with a new type of Bernstein-Durrmeyer operators on mobile interval. First, we represent the operators in terms of hypergeometric series. We also establish local and global approximation results for these operators in terms of modulus of continuity. We obtain an asymptotic formula for these operators and in the last section we present better error estimation for the operators using King type approach. (C) 2015 Mathematical Institute Slovak Academy of SciencesÖğe ON BASKAKOV OPERATORS PRESERVING THE EXPONENTIAL FUNCTION(Publishing House of the Romanian Academy, 2017) Yilmaz, Övgü Gürel; Gupta, Vijay; Aral, AliIn this paper, we are concerned about the King-type Baskakov operators defined by means of the preserving functions e0 and e2ax, a > 0 fixed. Using the modulus of continuity, we show the uniform convergence of the new operators to f. Also, by analyzing the asymptotic behavior of King-type operators with a Voronovskaya-type theorem, we establish shape preserving properties using the generalized convexity. © 2017, Publishing House of the Romanian Academy. All rights reserved.Öğe On q-Baskakov type operators(Walter de Gruyter GmbH, 2009) Aral, Ali; Gupta, VijayIn the present paper we introduce two g-analogous of the well known Baskakov operators. For the first operator we obtain convergence property on bounded interval. Then we give the montonity on the sequence of q-Baskakov operators for n when the function f is convex. For second operator, we obtain direct approximation property on unbounded interval and estimate the rate of convergence. One can say that, depending on the selection of q, these operators are more flexible then the classical Baskakov operators while retaining their approximation properties. © 2009 Warsaw University. All rights reserved.Öğe ON q-BASKAKOV TYPE OPERATORS(De Gruyter Poland Sp Z O O, 2009) Aral, Ali; Gupta, VijayIn the present paper we introduce two q-analogous of the well known Baskakov operators. For the first operator we obtain convergence property on bounded interval. Then we give the montonity on the sequence of q- Baskakov operators for n when the function f is convex. For second operator, we obtain direct approximation property on unbounded interval and estimate the rate of convergence. One can say that, depending on the selection of q, these operators are more flexible then the classical Baskakov operators while retaining their approximation properties.Öğe On semi-exponential Gauss-Weierstrass operators(Springer Basel Ag, 2022) Gupta, Vijay; Aral, Ali; Ozsarac, FiratThe paper deals with the semi-exponential type Gauss-Weierstrass operators. The central moments of these operators are constant functions. We estimate some direct results. We also provide a modification of such operators so as the preserve two exponential functions, we estimate weighted approximation, quantitative asymptotic formula for the modified form of operators. Later, a comparison is stated, that shows that in a certain sense and for certain family of illustrative functions the modified form of the semi-exponential Gauss-Weierstrass operators approximates better than the classical ones. Finally, we give some numerical results, which help us better understand the theoretical results obtained in the previous sections.Öğe On semi-exponential Gauss–Weierstrass operators(Birkhauser, 2022) Gupta, Vijay; Aral, Ali; Özsaraç, FiratThe paper deals with the semi-exponential type Gauss–Weierstrass operators. The central moments of these operators are constant functions. We estimate some direct results. We also provide a modification of such operators so as the preserve two exponential functions, we estimate weighted approximation, quantitative asymptotic formula for the modified form of operators. Later, a comparison is stated, that shows that in a certain sense and for certain family of illustrative functions the modified form of the semi-exponential Gauss–Weierstrass operators approximates better than the classical ones. Finally, we give some numerical results, which help us better understand the theoretical results obtained in the previous sections. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.Öğe On the Durrmeyer type modification of the q-Baskakov type operators(Pergamon-Elsevier Science Ltd, 2010) Aral, Ali; Gupta, VijayThis paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [0, infinity). An estimate for the rate of convergence and weighted approximation properties are also obtained. (C) 2009 Elsevier Ltd. All rights reserved.Öğe On the modification of the Szasz-Durrmeyer operators(Walter De Gruyter Gmbh, 2016) Aral, Ali; Deniz, Emre; Gupta, VijayIn this paper we consider the modification of Szasz-Durrmeyer operators based on the Jain basis function. Voronovskaya-type estimates of point-wise convergence along with its quantitative version based on the weighted modulus of smoothness are given. Moreover, a direct approximation theorem for the operators is proved.Öğe On the q analogue of Stancu-Beta operators(Pergamon-Elsevier Science Ltd, 2012) Aral, Ali; Gupta, VijayIn the present work we introduce the q analogue of well known Stancu-Beta operators. We estimate moments and establish direct results in terms of the modulus of continuity. We also present an asymptotic formula. (C) 2011 Elsevier Ltd. All rights reserved.
