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We introduce a genuine summation-integral type operators based on Lupas-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and K-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove a Korovkin type approximation theorem and obtain the rate of convergence of the generalized Szász–Mirakjan–Beta operators by means of modulus of continuity and elements of Lipschitz class. Furthermore we give the -statistical approximation theorem for these operators and investigate the case which provides the best estimation. [Copyright &y& Elsevier]

In this paper, we obtain a direct result in local approximation for the operator introduced by W. Li [5]. For a modified version of Li's operator, we establish a direct local and global approximation theorem. We also estimate the rate of approximation for the Beziér variant of Li's operator. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we consider the Bezier variant of the generalized Durrmeyer type operators, and obtain an estimate on the rate of convergence of for the decomposition technique of functions of bounded variation. In the end we propose an open problem for the readers and give an asymptotic formula for these generalized Durrmeyer type operators. [Copyright &y& Elsevier]

Abstract: In this paper we study the mixed summation-integral type operators having Szász and Beta basis functions. We extend the study of Gupta and Noor [V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szász–Beta operators, J. Math. Anal. Appl. 321 (1) (2006) 1–9] and obtain some direct results in local approximation without and with iterative combinations. In the last section are established direct global approximation theorems. [Copyright &y& Elsevier]

Abstract: In this paper we study the mixed summation–integral type operators having Szasz and Beta basis functions in summation and integration, respectively, we obtain the rate of point-wise convergence, a Voronovskaja type asymptotic formula and an error estimate in simultaneous approximation. [Copyright &y& Elsevier]

Abstract: Srivastava and Gupta [H.M. Srivastava, V. Gupta, A certain family of summation integral type operators, Math. Comput. Modelling 37 (2003) 1307–1315] proposed a certain family of summation integral type operators and estimated the rate of convergence for bounded variation functions. Very recently Gupta et al. [V. Gupta, R.N. Mohapatra, Z. Finta, On certain family of mixed summation-integral type operators, Math. Comput. Modelling, in press] defined the mixed summation integral type operators and obtained some direct results in simultaneous approximation. In the present paper, we consider the linear combinations of the mixed summation integral type operators and establish the error estimation for simultaneous approximation in terms of higher order modulus of continuity. To prove the main result we use the technique of linear approximating method viz. Steklov mean. [Copyright &y& Elsevier]