Some types of convergence and its applications to approximation theory

This thesis consists of six chapters. The first chapter of the thesis is the introduction chapter and it includes historical background of the subject and also a brief overview of the study. In the second chapter, some basic definitions and the concepts of statistical convergence and ideal convergence have been recalled. The original results of the thesis have been given in the third, fourth, fifth and sixth chapters. In the third chapter, the concepts of (S_N̅)(I)-convergence and I-(N̅,p_n)-summability have been introduced and also some relations between these concepts have been examined. In the fourth chapter, using the concept of I-(N̅,p_n)-summability, classical Korovkin approximation theorem has been obtained in more general form. In fifth chapter, weighted statistical convergence ((S_(N̅_(m,n)))-convergence) and summability method for double sequence have been introduced, and the relations between these methods have been examined and also the relations between (S_(N̅_(m,n)))-convergence and the concepts of statistical convergence and Riesz convergence have been studied. In the final chapter, some new sequence spaces have been defined through a modulus function and some inclusion relations of these sequence spaces have been investigated.