DEMONSTRATIO MATHEMATICA Vol. XLII No 3 2009 Prema Maheshwari (Sharma) DIRECT THEOREMS FOR MODIFIED BASKAKOV OPERATORS IN Lp-SPACES Abstract. In the year 1993, Gupta and Srivastava [3] introduced the integral mod- ification of the well known Baskakov operators by taking the weight functions of Szasz basis function, so called Baskakov-Szasz operators. In this paper, we obtain some direct theorems for the linear combination of these Baskakov Szasz type operators. To prove our one of the direct theorems, we use the technique of a mathematical tool which is the linear approximating method and is known as the Steklov means. 1. Introduction For / e L p [0, oo), p > 1, Gupta and Srivastava [3] introduced an interest- ing sequence of linear positive operators to modify the well-known Baskakov operators by considering the weights of Szasz basis functions. The modified Baskakov-Szasz operators, introduced in [3] are defined by OO oo (1.1) s n (f,x) =nJ2Pn,v(x) \ Qn,v{ u )f( u ) du i x e [0, oo), t)=0 o where (1.2) p n , v (x)=( n + V ~ 1 )x v (l + xr n - v and q n>v (u) = ^ ^ , V V J vl In [3], the authors have estimated asymptotic formula and an error esti- mates in simultaneous approximation. It is observed from [3] that the rate of convergence for these operators S n (f,x) is of O (n^ 1 ). To improve the order of approximation, we consider the linear combination of these opera- tors S n (f, k, x) of the operators Sd jn (f, x ), where djti, j = 0,1,2,..., k are Key words and phrases: Steklov mean, linear combination, linear positive operators, total variation. 1991 Mathematics Subject Classification: 41A25, 41A30. Unauthenticated Download Date | 7/28/18 5:53 AM