An Introductory Course in Summability Theory

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 various 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.

Preface

About the authors

About the book

Chapter 1: Introduction and General Matrix Methods

1.1 Brief Introduction

1.2 General Matrix Methods

1.3 Exercises

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 (C, 1) Method

2.4. Exercises

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 Exercises

References

Chapter 4: Tauberian Theorems

4.1 Brief Introduction

4.2 Tauberian Theorems

4.3 Exercises

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 Mx

5.4 Matrix transforms for reversible methods

5.5 Matrix transforms for normal methods

5.6 Exercises

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 Mx

6.4 Matrix transforms for regular perfect methods

6.5 Exercises

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 Exercises

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 Exercises

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 Exercises

References

Index

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.