# An Introductory Course in Summability Theory

## Books 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 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.

• Discusses results on matrix transforms of several matrix methods

• The only English-language 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 end-of-chapter 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.

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: Inverse-Transformation 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 (mλ, mμ), (cλ, cμ) and (cλ, mμ)

8.3 Matrix transforms from mAλ into mBμ

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 mλ, cλ, cAλ and mAλ

9.3 Matrix transforms from cAλ into cBμ or mBμ

9.4 Exercises

References

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