The approximation properties of king type generalizations of q-bernstein operators

Korovkin type approximation theorems have fundamental role in the approximation theory. Through 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 could be investigated. In this thesis, firstly King-type then Kantorovich-type generalizations of q-analogue of the Bernstein operators defined by Lupaş [Lupaş 2006] are given and statistical approximation properties of these generalizations are investigated. In the approximation theory, evaluation of approximation rate is as important subject as an investigation of uniform convergence of operators. In this thesis, King-type generalization of q-analogue of the Bernstein operator is firstly defined. It has been shown that, in terms of both the modulus of continuity and Lipschitz class functions, the rate of convergence of this operator is better then the rate of convergence of q-analogue of the Bernstein operators given by Lupaş [Lupaş 1987] under some conditions. This situation increases the importance of the thesis. The statistical approximation of King-type generalization of q-analogue of the Bernstein operator is also investigated. Furthermore, through investigating the statistical approximation properties, statistical approximation rate of this operator is found with the aid of modulus of continuity and Lipschitz class functions respectively. Moreover, first type Kantorovich-type generalization of q-analogue of the Bernstein operator is defined via q-integral and approximation properties are given. Then, second type Kantorovich-type generalization of q-analogue of the Bernstein operator is defined via Riemann type q-integral and the approximation properties of both operators are investigated. Lastly, statistical approximation rates of these operators are found with the aid of modulus of continuity and Lipschitz class functions and these rates are compared. It is observed that, approximation rate of the first type operator is better.