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Approximation by Modified Jain-Baskakov Operators  [PDF]
Vishnu Narayan Mishra,Preeti Sharma,Marius Birou
Mathematics , 2015,
Abstract: In the present paper we discuss the approximation properties of Jain-Baskakov operators with parameter c. The present paper deals with the modified forms of the Baskakov basis functions. We establish some direct results, which include the asymptotic formula and error estimation in terms of the modulus of continuity and weighted approximation.
On Rate of Approximation by Modified Beta Operators  [PDF]
Prerna Maheshwari (Sharma),Vijay Gupta
International Journal of Mathematics and Mathematical Sciences , 2009, DOI: 10.1155/2009/205649
Abstract: We establish the rate of convergence for the modified Beta operators (,), for functions having derivatives of bounded variation.
-inverse theorem for modified beta operators
Vijay Gupta,Prerna Maheshwari,V. K. Jain
International Journal of Mathematics and Mathematical Sciences , 2003, DOI: 10.1155/s0161171203210103
Abstract: We obtain a converse theorem for the linear combinations of modified beta operators whose weight function is the Baskakov operators. To prove our inverse theorem, we use the technique of linear approximating method, namely, Steklov mean.
Some Approximation Properties of -Baskakov-Beta-Stancu Type Operators  [PDF]
Vishnu Narayan Mishra,Kejal Khatri,Lakshmi Narayan Mishra
Journal of Calculus of Variations , 2013, DOI: 10.1155/2013/814824
Abstract: This paper deals with new type -Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of -integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the -operators. 1. Introduction In the recent years, the quantum calculus ( -calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of -calculus in the area of approximation theory, -generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1–3]). In this direction, several authors have proposed the -analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta [4] defined -generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta [5] obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval , modified Beta operators [6] are defined as where and . The discrete -Beta operators are defined as Recently, Maheshwari and Sharma [7] introduced the -analogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of -Baskakov operators, we propose the -analogue of the operators , recently introduced and studied for special values by Gupta and Kim [8] as where and . We know that and . We mention that (see [8]). Very recently, Gupta et al. [9] introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based on -integers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of -calculus. Such notations can be found in [10, 11]. We consider as a real number satisfying . For , The -binomial coefficients are given by The -derivative of a function is given by The -analogues of product and quotient rules are defined as The -Jackson integrals and the
On Simultaneous Approximation of Modified Baskakov Durrmeyer Operators  [PDF]
Prashantkumar Patel,Vishnu Narayan Mishra
Mathematics , 2015,
Abstract: In this present manuscript, we discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter $\gamma>0$. We compute the moments of these modified operators. Also, establish point-wise convergence, Voronovskaja type asymptotic formula and an error estimation in terms of second order modification of continuity of the function for the operators $B_{n,\gamma}^{\alpha,\beta}(f,x)$.
A Note on the Statistical Approximation Properties of the Modified Discrete Operators  [PDF]
Reyhan Canatan
Open Journal of Discrete Mathematics (OJDM) , 2012, DOI: 10.4236/ojdm.2012.23022
Abstract: In this present paper, firstly, the modified positive operators and their discrete operators are constructed. Then, we investigate the statistical approximation properties and rates of convergence by using modulus of continuity of these positive linear operators. Finally, we obtain the rate of statistical convergence of truncated operators.
Approximation properties of certain modified Szasz-Mirakyan operators
Lucyna Rempulska,Zbigniew Walczak
Le Matematiche , 2000,
Abstract: We introduce certain modified Szasz - Mirakyan operators in exponential weighted spaces of functions of one variable. We give theorems on the degree of approximation and the Voronovskaya type theorem.
Approximation theorems for modified Szasz-Mirakjan operators in polynomial weight spaces
Monika Herzog
Le Matematiche , 1999,
Abstract: In this paper we will study properties of Szasz-Mirakjan type operators A_n^ν , B_n^ ν defined by modified Bessel function I_ν . We shall present theorems giving a degree of approximation for these operators.
On strong approximation by modified Meyer-K"onig and Zeller operators
L. Rempulska,M. Skorupka
Tamkang Journal of Mathematics , 2006, DOI: 10.5556/j.tkjm.37.2006.123-130
Abstract: We introduce certain modified Meyer-K"onig and Zeller operators $ M_{n;r} $ in the space of $r $-th times differentiable functions $ f $ and we study strong differences $ H_{n;r}^q(f) $ for them. This note is motivated by results on strong approximation connected with Fourier series ([7]).
Approximation properties of modified Szász-Mirakyan operators in polynomial weighted space  [PDF]
Prashantkumar Patel,Vishnu Narayan Mishra,Mediha ?rkcü
Mathematics , 2015,
Abstract: We introduce certain modified Sz\'{a}sz-Mirakyan operators in polynomial weighted spaces of functions of one variable. We studied approximation properties of these operators.
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