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.

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.

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

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 $B_{n,\gamma}^{\alpha,\beta}(f,x)$.

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.

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.

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.

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]).

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.