Preface

This book is intended for graduate students and researchers who have interest in functional analysis, in general and summability theory, in particular. It describes several useful topics in summability theory along with applications. The book consists of nine chapters and is organized as follows: Chapter “An Introduction to Summability Methods” is introductory in nature. This chapter focuses on the historical development of summability theory right from Cauchy’s concept to till date. Summability methods developed from the two basic processes—T-process and /-process—have also been discussed in this chapter. Chapter “Some Topics in Summability Theory” deals with the study of some classical and modern summability methods, and the connections among them. In fact, results concerning summability by weighted mean method, the ðM; knÞ method, the Abel method, and the Euler method are presented. Then the sequence space Kr, r 1 being a fixed integer, is defined and a Steinhaus type theorem is proved. The space Kr is then studied in the context of sequences of 0’s and 1’s. Further, the core of a sequence is studied, an improvement of a result of Sherbakhoff is proved and a very simple proof of Knopp’s core theorem is then deduced. Finally, a study of the matrix class ð‘; ‘Þ is presented. Chapter “Summability and Convergence Using Ideals” is concentrated on different concepts of summability and convergence using the notions of ideals and essentially presents the basic developments of these notions. This chapter starts with the first notion of ideal convergence and goes on to discuss in detail how the notion has been extended over the years from single sequences to double sequences and nets. This chapter also discusses some of the most recent advances made in this area, in particular applications of ideal convergence to the theory of convergence of sequences of functions. Some problems are also listed which still remain open. In chapter “Convergence Acceleration and Improvement by Regular Matrices”, a new, non-classical convergence acceleration concept is studied and compared with the well-known classical convergence acceleration concept. It is shown that the new concept allows to compare the speeds of convergence for a larger set of