An Introductory Course in Summability Theory 1st Edition by Ants Aasma, Hemen Dutta, P N Natarajan ISBN 1119397782 9781119397786

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ISBN 10: 1119397782 
ISBN 13: 9781119397786
Author: Ants Aasma, Hemen Dutta, P N Natarajan

In creating this book, the authors’ intent was to provide graduate students, researchers, physicists, and engineers with a reasonable introduction to summability theory. Over the course of nine chapters, the authors cover all of the fundamental concepts and equations informing summability theory and its applications, as well as some of its lesser known aspects. Following a brief introduction to the history of summability theory, general matrix methods are introduced, and the Silverman-Toeplitz theorem on regular matrices is discussed. A variety of special summability methods, including the Nörlund method, the Weighted Mean method, the Abel method, and the (C, 1) – method are next examined. An entire chapter is devoted to a discussion of some elementary Tauberian theorems involving certain summability methods. Following this are chapters devoted to matrix transforms of summability and absolute summability domains of reversible and normal methods; the notion of a perfect matrix method; matrix transforms of summability and absolute summability domains of the Cesàro and Riesz methods; convergence and the boundedness of sequences with speed; and convergence, boundedness, and summability with speed.

An Introductory Course in Summability Theory 1st Table of contents:

Chapter 1: Introduction and General Matrix Methods
1.1 Brief Introduction
1.2 General Matrix Methods
1.3 Excercise
References
Chapter 2: Special Summability Methods I
2.1 The Nörlund Method
2.2 The Weighted Mean Method
2.3 The Abel Method and the Method
2.4 Excercise
References
Chapter 3: Special Summability Methods II
3.1 The Natarajan Method and the Abel Method
3.2 The Euler and Borel Methods
3.3 The Taylor Method
3.4 The Hölder and Cesàro Methods
3.5 The Hausdorff Method
3.6 Excercise
References
Chapter 4: Tauberian Theorems
4.1 Brief Introduction
4.2 Tauberian Theorems
4.3 Excercise
References
Chapter 5: Matrix Transformations of Summability and Absolute Summability Domains: Inverse-Transformation Method
5.1 Introduction
5.2 Some Notions and Auxiliary Results
5.3 The Existence Conditions of Matrix Transform
5.4 Matrix Transforms for Reversible Methods
5.5 Matrix Transforms for Normal Methods
5.6 Excercise
References
Chapter 6: Matrix Transformations of Summability and Absolute Summability Domains: Peyerimhoff’s Method
6.1 Introduction
6.2 Perfect Matrix Methods
6.3 The Existence Conditions of Matrix Transform
6.4 Matrix Transforms for Regular Perfect Methods
6.5 Excercise
References
Chapter 7: Matrix Transformations of Summability and Absolute Summability Domains: The Case of Special Matrices
7.1 Introduction
7.2 The Case of Riesz Methods
7.3 The Case of Cesàro Methods
7.4 Some Classes of Matrix Transforms
7.5 Excercise
References
Chapter 8: On Convergence and Summability with Speed I
8.1 Introduction
8.2 The Sets , , and
8.3 Matrix Transforms from into
8.4 On Orders of Approximation of Fourier Expansions
8.5 Excercise
References
Chapter 9: On Convergence and Summability with Speed II
9.1 Introduction
9.2 Some Topological Properties of , , and
9.3 Matrix Transforms from into or
9.4 Excercise

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