**Institution Of The Thesis:** Gazi University, Fen Bilimleri Enstitüsü, Turkey

**Approval Date:** 2011

**Thesis Language:** Turkish

**Student:** Mediha Örkcü

**Consultant: **OGÜN DOĞRU

Korovkin type approximation theorems have fundamental role in the approximation theory. By the aid of the concept of statistical approximation, Korovkin type theorems are developed [Gadjiev and Orhan, 2002; Duman et all, 2003]. Using these theorems, the approximation properties of the q − generalizations and integral-type generalizations of many linear positive operators can be investigated. In this thesis, Kantorovich-type generalization of q − Szasz-Mirakjan operators defined by Aral and Gupta (2006) is given and statistical approximation properties of this generalization are investigated. In the approximation theory, evaluation of approximation rate is as important subject as an investigation of uniform convergence of operators. Hence, Voronovskaja type theorem is given for constructed operators' sequence in this thesis. Then, a new Kantorovich type generalization of the q − Szasz-Mirakjan operators is defined to obtain statistical approximation rate by means of modulus of continuity. Furthermore, q − Szasz-Mirakjan-Kantorovich operators are developed with the aid of Riemann type q − integral and the approximation rate is obtained by the weighted modulus of continuity. Also, statistical approximation properties of two variable generalizations of operators constructed by the Riemann type q − integral are given. In the case of q = 1 , since our operators are reduced to the integral type generalization of the classical operators, and since the approximation rate can properly be calibrated by a proper choose of ( ) n q instead of q , increase the importance of this thesis.