Summability Theory and Its Applications explains various aspects of summability and demonstrates its applications in a rigorous and coherent manner. The content can readily serve as a reference or as a useful series of lecture notes on the subject. This substantially revised new edition includes brand new material across several chapters as well as several corrections, including: the addition of the domain of Cesaro matrix C(m) of order m in the classical sequence spaces to Chapter 4; and introducing the domain of four-dimensional binomial matrix in the spaces of bounded, convergent in the Pringsheim's sense, both convergent in the Pringsheim's sense and bounded, and regularly convergent double sequences, in Chapter 7. Features Investigates different types of summable spaces and computes their dual Suitable for graduate students and researchers with a (special) interest in spaces of single and double sequences, matrix transformations and domains of triangle matrices Can serve as a reference or as supplementary reading in a computational physics course, or as a key text for special Analysis seminars.

Summability Theory and Its Applications explains various aspects of summability and demonstrates its applications in a rigorous and coherent manner. The content can readily serve as a reference or as a useful series of lecture notes on the subject. This substantially revised new edition includes brand new material across several chapters as well as several corrections, including: the addition of the domain of Cesaro matrix C(m) of order m in the classical sequence spaces to Chapter 4; and introducing the domain of four-dimensional binomial matrix in the spaces of bounded, convergent in the Pringsheim's sense, both convergent in the Pringsheim's sense and bounded, and regularly convergent double sequences, in Chapter 7. Features Investigates different types of summable spaces and computes their dual Suitable for graduate students and researchers with a (special) interest in spaces of single and double sequences, matrix transformations and domains of triangle matrices Can serve as a reference or as supplementary reading in a computational physics course, or as a key text for special Analysis seminars.

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.

This book is aimed at both experts and non-experts with an interest in getting acquainted with sequence spaces, matrix transformations and their applications. It consists of several new results which are part of the recent research on these topics. It provides different points of view in one volume, e.g. their topological properties, geometry and summability, fuzzy valued study and more. This book presents the important role sequences and series play in everyday life, it covers geometry of Banach Sequence Spaces, it discusses the importance of generalized limit, it offers spectrum and fine spectrum of several linear operators and includes fuzzy valued sequences which exhibits the study of sequence spaces in fuzzy settings. This book is the main attraction for those who work in Sequence Spaces, Summability Theory and would also serve as a good source of reference for those involved with any topic of Real or Functional Analysis.

Summability is a mathematical topic with a long tradition and many applications in, for example, function theory, number theory, and stochastics. It was originally based on classical analytical methods, but was strongly influenced by modern functional analytical methods during the last seven decades. The present book aims to introduce the reader to the wide field of summability and its applications, and provides an overview of the most important classical and modern methods used. Part I contains a short general introduction to summability, the basic classical theory concerning mainly inclusion theorems and theorems of the Silverman-Toeplitz type, a presentation of the most important classes of summability methods, Tauberian theorems, and applications of matrix methods. The proofs in Part I are exclusively done by applying classical analytical methods. Part II is concerned with modern functional analytical methods in summability, and contains the essential functional analytical basis required in later parts of the book, topologization of sequence spaces as K- and KF-spaces, domains of matrix methods as FK-spaces and their topological structure. In this part the proofs are of functional analytical nature only. Part III of the present book deals with topics in summability and topological sequence spaces which require the combination of classical and modern methods. It covers investigations of the constistency of matrix methods and of the bounded domain of matrix methods via Saks space theory, and the presentation of some aspects in topological sequence spaces. Lecturers, graduate students, and researchers working in summability and related topics will find this book a useful introduction and reference work.

There are excellent books on both functional analysis and summability. Most of them are very terse. In Functional Analysis and Summability, the author makes a sincere attempt for a gentle introduction of these topics to students. In the functional analysis component of the book, the Hahn–Banach theorem, Banach–Steinhaus theorem (or uniform boundedness principle), the open mapping theorem, the closed graph theorem, and the Riesz representation theorem are highlighted. In the summability component of the book, the Silverman–Toeplitz theorem, Schur’s theorem, the Steinhaus theorem, and the Steinhaus-type theorems are proved. The utility of functional analytic tools like the uniform boundedness principle to prove some results in summability theory is also pointed out. Features A gentle introduction of the topics to the students is attempted. Basic results of functional analysis and summability theory and their applications are highlighted. Many examples are provided in the text. Each chapter ends with useful exercises. This book will be useful to postgraduate students, pre-research level students, and research scholars in mathematics. Students of physics and engineering will also find this book useful since topics in the book also have applications in related areas.

Incorporating substantial developments from the last thirty years into one resource, Asymptotics and Borel Summability provides a self-contained introduction to asymptotic analysis with special emphasis on topics not covered in traditional asymptotics books. The author explains basic ideas, concepts, and methods of generalized Borel summability, tr