Approximation with bivariate generalized linear positive operators

This thesis consists of seven chapters. The first chapter is devoted to introduction and basic theorems. In the second chapter, linear positive operators will be introduced and their basic properties will be analyzed. In the second chapter, the basic approximation properties of the Bernstein-Schurer and Gadjiev-Ibragimov operators, which are two important operators in the design of the thesis, will be given by stating that their one and two dimensional forms satisfy Korovkin or Volkov theorem. In the third part of the thesis, it will be shown that the generalized two-dimensional Bernstein-Schurer and Kantorovich generalization on a variably bounded domain satisfies the conditions of Volkov's theorem and the approximation speed will be calculated with the help of the continuity module and Lipschitz class functions. In the fourth part of the chapter, important approximation properties of the Gadjiev-Ibragimov and Kantorovich generalizations on generalized mobile intervals are first given and then the two-dimensional forms of the operators on variable bounded domains are given. Moreover, in the last part of this chapter, a generalization of the Gadjiev-Ibragimov operator to Fibonacci sequences is constructed and important approximation properties are presented. In the fourth section, it is shown that the defined operators satisfy the conditions of Korovkin or Volkov's theorem and the approximation rates of the operator are analyzed in terms of the modulus of continuity and functions of Lipschitz class. In the fifth chapter, it is aimed to give approximations of Bernstein-Schurer type operators with methods that are very new to the literature. The fifth chapter consists of six sections. In the first two parts, the α-modification of the one-dimensional Bernstein-Schurer and Kantorovich generalization, in the third section the approximation of conics by Bernstein-Schurer operators in two variables with double indices, in the fourth part the properties of the approximation by Bernstein-Schurer polynomials of rational type, in the fifth section the Riemann-Liouville type generalization of fractional Bernstein-Schurer-Kantorovich operators of order α, and in the last part a Balazs-type modification of two-dimensional Bernstein-Schurer operators are studied. In the sixth chapter, Bernstein-type operators based on more than one representation parameter are defined and the necessary case analysis for preserving the positivity of the operator to satisfy the Korovkin-type theorem is studied. In the last chapter, conclusions and recommendations obtained from the thesis are presented.