Equi-statistical convergence

This thesis consists of three chapters. The first chapter is devoted to the introduction. The second chapter constitutes the part where the fundamental definitions and theorems are presented. In this chapter, firstly, statistical convergence is defined based on the concept of density, and then the concept of equi-statistical convergence, which includes the classical uniform convergence and statistical uniform convergence, is defined. Furthermore, Korovkin-type approximation theorems are given using these concepts. Afterwards, the concepts of lacunary statistical convergence and λ-statistical convergence are included, and based on these definitions, lacunary equi-statistical convergence and λ-equi-statistical convergence are introduced, and Korovkin-type theorems are given with the help of these concepts. In the third chapter, the power series method, which includes the Abel and Borel summability methods, is introduced and then statistical convergence in the sense of the power series method is defined, and it is explained with examples that this concept cannot be compared with statistical convergence. In addition, for sequences of functions, the concepts of statistical pointwise convergence, statistical uniform convergence, and equi-statistical convergence in the sense of the power series method are introduced. Afterwards, a Korovkin-type approximation theorem is obtained by means of the concept of equi-statistical convergence in the sense of the power series method. Moreover, the rate of P_p-equi-statistical convergence is obtained via the modulus of continuity is given, and finally, a Voronovskaya-type approximation theorem obtained using equi-statistical convergence in the sense of the power series method.