Matrix Summability Methods

The purpose of this study is to prepare a brief summary of some matrix summability methods. This thesis consists of four chapters. In the first chapter, a general introduction to the matrix summability method is mentioned. In Chapter 2, deals with the basic definitions and theorems of sequences associated with matrix summability methods. In this thesis contains some proper examples and definitions that are related to sequences, subsequences, bounded sequences, monotone sequences, convergent sequences, divergent sequences and Cauchy sequences. In addition to this, some basic theorems about sequences and basic properties of infinite matrices are given. Some of properties of infinite matrices; product of matrices, triangle matrices, inverse of matrices, triangular matrices. In Chapter 3, contains the theory of matrix summability methods. In the first part, basic definitions and theorems of matrix summability methods are examined. Then, these two theorems and their examples are given which enable us to learn whether the matrices are conservative matrix (Schur-Kojima Theorem) or regular matrix (Silverman-Teoblitz Theorem). Furthermore, comparable matrix methods (stronger or weaker matrix) are included. In the last section, zero preserving matrix, multiplicative matrix and some related theorems are examined. In the last chapter, some matrix summability methods are discussed together with general definitions, theorems, and examples. Also the proofs of the theorems are given. These methods are: Cesaro methods, Hölder methods, Euler Knopp methods, and Hausdorff matrix methods. Keywords: matrix summability methods, conservative matrix, regular matrix, Cesaro methods, Hausdorff methods, Riesz methods, Hölder methods, Euler methods.