An Introductory Course in Summability Theory

✍ Scribed by Aasma, Ants; Dutta, Hemen; Natarajan, Pinnangudi Narayanasubramanian

• Publisher: Wiley
• Year: 2017
• Tongue: English
• Leaves: 208
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Content: Preface ixAbout the Authors xiAbout the Book xiii1 Introduction and General Matrix Methods 11.1 Brief Introduction 11.2 General Matrix Methods 21.3 Exercise 16References 192 Special Summability Methods I 212.1 The Noerlund Method 212.2 The Weighted Mean Method 292.3 The Abel Method and the (C,1) Method 342.4 Exercise 44References 453 Special Summability Methods II 473.1 The Natarajan Method and the Abel Method 473.2 The Euler and Borel Methods 533.3 The Taylor Method 593.4 The Hoelder and Cesaro Methods 623.5 The Hausdorff Method 643.6 Exercise 73References 744 Tauberian Theorems 754.1 Brief Introduction 754.2 Tauberian Theorems 754.3 Exercise 83References 845 Matrix Transformations of Summability and Absolute Summability Domains: Inverse-Transformation Method 855.1 Introduction 855.2 Some Notions and Auxiliary Results 875.3 The Existence Conditions of Matrix Transform Mx 915.4 Matrix Transforms for Reversible Methods 955.5 Matrix Transforms for Normal Methods 1025.6 Exercise 107References 1096 Matrix Transformations of Summability and Absolute Summability Domains: Peyerimhoff s Method 1136.1 Introduction 1136.2 Perfect Matrix Methods 1136.3 The Existence Conditions of Matrix Transform Mx 1176.4 Matrix Transforms for Regular Perfect Methods 1216.5 Exercise 127References 1297 Matrix Transformations of Summability and Absolute Summability Domains: The Case of Special Matrices 1317.1 Introduction 1317.2 The Case of Riesz Methods 1317.3 The Case of Cesaro Methods 1397.4 Some Classes of Matrix Transforms 1487.5 Exercise 151References 154 8 On Convergence and Summability with Speed I 8.1 Introduction 8.2 The sets (m , m ), (c , c ) and (c , m ) 8.3 Matrix transforms from mA into mB 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 , c , cA and mA 9.3 Matrix transforms from cA into cB or mB 9.4 Exercises References

✦ Subjects

Sequences (Mathematics) / fast / (OCoLC)fst01112884.;Summability theory / fast / (OCoLC)fst01138488.;MATHEMATICS / Calculus / bisacsh.;MATHEMATICS / Mathematical Analysis / bisacsh.;Textbooks.;Sequences (Mathematics);Summability theory.;MATHEMATICS / Calculus.;MATHEMATICS / Mathematical Analysis.;Textbooks / fast / (OCoLC)fst01423863