**Thesis Type:** Postgraduate

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

**Approval Date:** 2010

**Student:** HAKAN DELİBAŞ

**Supervisor: **İSMET YÜKSEL

This master thesis consists of four chapters. In the first chapter, definition of linear positive operators is given and its fundamental properties are examined. In the second chapter, the definitions of q-calculus of some mathematical expressions are given and also theorems. In the third chapter, the fundamental properties of q-Bernstein operators are examined, and also this chapter, the definition of q-Durrmeyer operators is given and using the Korovkin theorem, its approximation properties are explained. In the fourth chapter, rate of approximation of q-Durrmeyer operators are examined with help of modulus of continuity, K-functionel, the Lipschitz type maximal function. Finally five chapter, the definition of D∞,q operators is given and its approximation properties are examined.