I -approximation properties of certain class of linear positive operators N.I. Mahmudov, M.A. zarslan and P. Sabanc‹gil Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin 10, Turkey Email: nazim.mahmudov@emu.edu.tr mehmetali.ozarslan@emu.edu.tr pembe.sabanc‹gil@emu.edu.tr Abstract In this paper we study I -approximation properties of certain class of linear positive operators. The two main tools used in this paper are I -convergence and Ditzian-Totik modulus of smoothness. Furthermore, we dene q-Lupa‚s-Durrmeyer operators and give local and global approximation results for such operators. 1 Introduction Let A := [a jn ]; j; n =1; 2; :::; be an innite summability matrix. The A-transform of x := fx n g n2N is the sequence Ax := f(Ax) j g j2N such that, (Ax) j := P 1 n=1 a jn x n , for every j 2 N. A summability matrix A is said to be regular (see [18]) if for every x =(x n ) for which lim n x = L we get lim j (Ax) j = L: A-statistical convergence [9], which is the generalization of the statistical convergence (see [?],[41]), is based on the concept of Adensity of the subset K  N: The Adensity of a subset K of N is given by  A (K) := lim j P n2K a j;n ; whenever the limit exists. A sequence x =(x n ) is said to be Astatistically convergent to L if  A fn 2 N : jx n  Lj "g =0; and it is denoted by st A  lim x = L (see[9], [10], [32]). For A = C 1 ; the Cesaro matrix of order one, C 1 -statistical convergence is known as the statistical convergence ([12],[11]). If A = I; the identity matrix, then I -statistical convergence is the ordinary convergence. It was proved by Kolk [25] that A-statistical convergence is stronger than ordinary convergence if lim j max n ja j;n j =0: Statistical convergence was rst used in approximation theory in [13]. Very recently, a new concept of convergence which is called I convergence, has been the subject of many mathematicians since it has advantages that it relaxes some conditions on the ordinary convergence. I convergence is based on the ideal of subsets of N: A collection I of subsets of N is said to be an ideal in N if the following conditions are satised: (i)  2I ; (ii) A; B 2I = ) A [ B 2I (iii) A 2I ;B  A = ) B 2I : An ideal is called admissible if the singleton fng2I for all n 2 N. Let I be a nontrivial ideal in N; i.e. N = 2I , then a sequence x := (x n ) is said to be Iconvergent to x if for every "> 0; fn : jx n  Lj "g2I (see [26] [27]). If I = fK  N :  A (K)=0g ; then 1