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

**Approval Date:** 2010

**Thesis Language:** Turkish

**Student:** Hüseyin Erhan ALTIN

**Consultant: **OGÜN DOĞRU

This master thesis consist of six chapters. In the first chapter, some definitions of a sequence of linear positive operators are given and their fundamental properties are obtained. The Korovkin theorem and its prof is also given. In the second chapter, definition of q − Bernstein polynomials is given and its approximation properties are obtained with the help of Korovkin theorem. The third chapter is devoted to the estimation of order of approximation of the q − Bernstein polynomials to the function f with the help of modulus of continutiy and K- functionals of Peetre. Also, the monotoncity properties of q − Bernstein polynomials are investigated when f is a convex function. Moreover, the estimation of speed of approximation of the q − Bernstein polynomials to the function f is obtained for the functions in Lipschitz class. In the fourth chapter, King type generalizations of q − Bernstein polynomials are introduced and the approximation properties of these polynomials are obtained like third chapter.In the fifth chapter, the Volkov theorem is given, then the definition of bivariate q − Bernstein poliynomials is given and their approximation properties are examined with the help of Volkov theorem. In the last chapter, King type bivariate generalizations of q − Bernstein polynomials are introduced and their aproximation properties are studied with the help of Volkov theorem.