Absolute Riesz and Related Summability Methods

Abstract

This chapter contains seven sections. The first section deals with historical developments of summability theory and some features of summability methods. The second section is devoted to generalized absolute Cesaro summability. In this section, some theorems on \(\varphi -{|C,1|}_k\), \(\varphi -{|{C,\alpha }|}_k\) and \(\varphi -{|{C,1;\delta }|}_k\) have been given. The third section is about some applications of quasi monotone sequences. In this section, some well-known theorems concerning the absolute Riesz summability and absolute matrix summability have been given. The fourth section is devoted to some applications of absolute matrix summability using almost increasing sequences. The fifth section deals with \(\varphi -\left| A,p_{n}\right| _{k}\) summability. In this section, a main theorem on absolute Riesz summability and its generalization to \(\varphi -\left| A,p_{n}\right| _{k}\) summability have been given. The sixth section deals with some applications of quasi power increasing sequences. Finally, the seventh section is related to generalized absolute Riesz and absolute matrix summability methods of infinite series.