Error estimation for mixed summation-integral type operators

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.