Some Schurer Type q-Bernstein Operators

ABSTRACT: In this thesis consist of six chapters. The introduction is given in the first chapter. In the second chapter, some necessary definitions, preliminaries and theorems are given. In this chapter, we also give the important theorems; by Korovkin and Volkov, Bernstein polynomials in one two variables, q-Bernstein, Bernstein-Chlodowsky and q-Bernstein Chlodowsky polynomials. In the third chapter, q-Bernstein Schurer operators are defined. Many properties and results of these polynomials, such as Korovkin type approximation and the rate of convergence of these operators in terms of Lipschitz class functional are given. In the fourth chapter q-Bernstein-Schurer-Chlodowsky operators are introduced. Korovkin type approximation theorem is given and the rate of convergence of this approximation is obtained by means of modulus of continuity of the function is obtained. In the fifth chapter, Schurer-type q-Bernstein Kantorovich operators are defined. Moreover the order of convergence of the operators in terms of modulus of continuity of the derivative of the function, and elements of Lipschitz classes are discussed. In the last chapter, Kantorovich type q-Bernstein operators are defined. Furthermore, Korovkin type approximation theorem is proved and the rate of convergence of this approximation are given. Keywords: q-Bernstein Schurer operators, Korovkin theorem, Schurer Type q-Bernstein Polynomials, Kantorovich type q-Bernstein-Schurer-Chlodovsky operators. ……………………………………………………………………………………………………………………………………………………………………………………………………………………