An Introductory Course in Summability TheoryISBN: 9781119397694
224 pages
April 2017

Description
An introductory course in summability theory for students, researchers, physicists, and engineers
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 SilvermanToeplitz 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.
• Discusses results on matrix transforms of several matrix methods
• The only Englishlanguage textbook describing the notions of convergence, boundedness, and summability with speed, as well as their applications in approximation theory
• Compares the approximation orders of Fourier expansions in Banach spaces by different matrix methods
• Matrix transforms of summability domains of regular perfect matrix methods are examined
• Each chapter contains several solved examples and endofchapter exercises, including hints for solutions
An Introductory Course in Summability Theory is the ideal first text in summability theory for graduate students, especially those having a good grasp of real and complex analysis. It is also a valuable reference for mathematics researchers and for physicists and engineers who work with Fourier series, Fourier transforms, or analytic continuation.
ANTS AASMA, PhD, is Associate Professor of Mathematical Economics in the Department of Economics and Finance at Tallinn University of Technology, Estonia.
HEMEN DUTTA, PhD, is Senior Assistant Professor of Mathematics at Gauhati University, India.
P.N. NATARAJAN, PhD, is Formerly Professor and Head of the Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, Tamilnadu, India.
Table of Contents
Preface ix
About the Authors xi
About the Book xiii
1 Introduction and General Matrix Methods 1
1.1 Brief Introduction 1
1.2 General Matrix Methods 2
1.3 Exercise 16
References 19
2 Special Summability Methods I 21
2.1 The Nörlund Method 21
2.2 The Weighted Mean Method 29
2.3 The Abel Method and the (C,1) Method 34
2.4 Exercise 44
References 45
3 Special Summability Methods II 47
3.1 The Natarajan Method and the Abel Method 47
3.2 The Euler and Borel Methods 53
3.3 The Taylor Method 59
3.4 The Hölder and Cesàro Methods 62
3.5 The Hausdorff Method 64
3.6 Exercise 73
References 74
4 Tauberian Theorems 75
4.1 Brief Introduction 75
4.2 Tauberian Theorems 75
4.3 Exercise 83
References 84
5 Matrix Transformations of Summability and Absolute Summability Domains: InverseTransformation Method 85
5.1 Introduction 85
5.2 Some Notions and Auxiliary Results 87
5.3 The Existence Conditions of Matrix Transform Mx 91
5.4 Matrix Transforms for Reversible Methods 95
5.5 Matrix Transforms for Normal Methods 102
5.6 Exercise 107
References 109
6 Matrix Transformations of Summability and Absolute Summability Domains: Peyerimhoff’s Method 113
6.1 Introduction 113
6.2 Perfect Matrix Methods 113
6.3 The Existence Conditions of Matrix Transform Mx 117
6.4 Matrix Transforms for Regular Perfect Methods 121
6.5 Exercise 127
References 129
7 Matrix Transformations of Summability and Absolute Summability Domains: The Case of Special Matrices 131
7.1 Introduction 131
7.2 The Case of Riesz Methods 131
7.3 The Case of Cesàro Methods 139
7.4 Some Classes of Matrix Transforms 148
7.5 Exercise 151
References 154
8 On Convergence and Summability with Speed I
8.1 Introduction
8.2 The sets (m^{λ}, m^{μ}), (c^{λ}, c^{μ}) and (c^{λ}, m^{μ})
8.3 Matrix transforms from m_{A}^{λ} into m_{B}^{μ}
8.4 On orders of approximation of Fourier expansions
8.5 Exercises
References
9 On Convergence and Summability with Speed II
9.1 Introduction
9.2 Some topological properties of m^{λ}, c^{λ}, c_{A}^{λ} and m_{A}^{λ}
9.3 Matrix transforms from c_{A}^{λ} into c_{B}^{μ} or m_{B}^{μ}
9.4 Exercises
References
Author Information
Ants Aasma, PhD, is Associate Professor of Mathematical Economics in the Department of Economics and Finance at Tallinn University of Technology, Estonia.
Hemen Dutta, PhD, is Senior Assistant Professor of Mathematics at Gauhati University, India.
P.N. Natarajan, PhD, is Formerly Professor and Head of the Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, Tamilnadu, India.