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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Table of Contents
Cover
Title Page
Copyright
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 Excercise
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 Method
2.4 Excercise
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 Excercise
References
Chapter 4: Tauberian Theorems
4.1 Brief Introduction
4.2 Tauberian Theorems
4.3 Excercise
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
5.4 Matrix Transforms for Reversible Methods
5.5 Matrix Transforms for Normal Methods
5.6 Excercise
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
6.4 Matrix Transforms for Regular Perfect Methods
6.5 Excercise
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 Excercise
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 Excercise
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 Excercise
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
An Introductory Course in Summability Theory
Ants Aasma
Hemen Dutta
P.N. Natarajan
This edition first published 2017
© 2017 John Wiley & Sons, Inc.
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Library of Congress Cataloguing-in-Publication Data
Names: Aasma, Ants, 1957- | Dutta, Hemen, 1981- | Natarajan, P. N., 1946
Title: An introductory course in summability theory Ants Aasma, Hemen Dutta, P.N. Natarajan.
Other titles: Introductory course in summability theory
Description: Hoboken, NJ : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2017004078 (print) | LCCN 2017001841 (ebook) | ISBN 9781119397694 (cloth) | ISBN 9781119397731 (Adobe PDF) | ISBN 9781119397779 (ePub)
Subjects: LCSH: Summability theory–Textbooks. | Sequences (Mathematics)–Textbooks.
Classification: LCC QA292 .A27 2017 (ebook) | LCC QA292 (print) | DDC 515/.243–dc23
LC record available at https://lccn.loc.gov/2017004078
Cover image: © naqiewei/Gettyimages
Cover design by Wiley
Preface
This book is intended for graduate students and researchers as a first course in summability theory. The book is designed as a textbook as well as a reference guide for students and researchers. Any student who has a good grasp of real and complex analysis will find all the chapters within his/her reach. Knowledge of functional analysis will be an added asset. Several problems are also included in the chapters as solved examples and chapter-end exercises along with hints, wherever felt necessary. The book consists of nine chapters and is organized as follows:
In chapter 1, after a very brief introduction to summability theory, general matrix methods are introduced and Silverman–Toeplitz theorem on regular matrices is proved. Schur's, Hahn's, and Knopp–Lorentz theorems are then taken up. Steinhaus theorem that a matrix cannot be both regular and a Schur matrix is then deduced.
Chapter 2 is devoted to a study of some special summability methods. The Nörlund method, the Weighted Mean method, the Abel method, and the (C; 1) method are introduced, and their properties are discussed.
Chapter 3 is devoted to a study of some more special summability methods. The (M; λn) method, the Euler method, the Borel method, the Taylor method, the Hölder and Cesàro methods, and the Hausdorff method are introduced, and their properties are discussed.
In Chapter 4, various Tauberian theorems involving certain summability methods are discussed.
In Chapter 5, matrix transforms of summability and absolute summability domains of reversible and normal methods are studied. The notion of M-consistency of matrix methods A and B is introduced and its properties are studied. As a special case, some inclusion problems are analyzed.
In Chapter 6, the notion of a perfect matrix method is introduced. Matrix transforms of summability domains of regular perfect matrix methods are considered.
In Chapter 7, matrix transforms of summability and absolute summability domains of the Cesàro and the Riesz methods are studied. Also, some special classes of matrices transforming the summability or absolute summability domain of a matrix method into the summability or absolute summability domain of another matrix method are considered.
In Chapter 8, the notions of the convergence and the boundedness of sequences with speed λ (λ is a positive monotonically increasing sequence) are introduced. The necessary and sufficient conditions for a matrix A to be transformed from the set of all λ-bounded or λ-convergent sequences into the set of all λ-bounded or μ-convergent sequences (μ is another speed) are described. In addition, the notions of the summability and the boundedness with speed by a matrix method are introduced and their properties are described. Also, the M-consistency of matrix methods A and B on the set of all sequences, λ-bounded by A, is investigated. As applications of main results, the matrix transforms for the case of Riesz methods are investigated, and the comparison of approximation orders of Fourier expansions in Banach spaces by different matrix methods is studied.
Chapter 9 continues the investigation of convergence, boundedness, and summability with speed, started in Chapter 8. Some topological properties of the spaces mλ (the set of all λ bounded sequences), cλ (the set of all λ-convergent sequences), cλA (the set of all sequences, λ- convergent by A), and mλA (the set of all sequences, λ-bounded by A) are introduced. The notions of λ-reversible, λ-perfect, and λ-conservative matrix methods are introduced. The necessary and sufficient conditions for a matrix M to be transformed from cλA into cλB or into mμB are described. Also, the M-consistency of matrix methods A and B on cλA is investigated. As applications of main results, the matrix transforms for the cases of Riesz and Cesàro methods are investigated.
We were influenced by the work of several authors during the preparation of the text. Constructive criticism, comments, and suggestions for the improvement of the contents of the book are always welcome. The authors are thankful to several researchers and colleagues for their valuable suggestions. Special thanks to Billy E. Rhoades, emeritus professor, Indiana University, USA, for editing the final draft of the book.
Ants Aasma, Tallinn, EstoniaHemen Dutta, Guwahati, IndiaP.N. Natarajan, Chennai, IndiaDecember, 2016
About the Authors
Ants Aasma is an associate professor of mathematical economics in the department of economics and finance at Tallinn University of Technology, Estonia. He received his PhD in mathematics in 1993 from Tartu University, Estonia. His main research interests include topics from the summability theory, such as matrix methods, matrix transforms, summability with speed, convergence acceleration, and statistical convergence. He has published several papers on these topics in reputable journals and visited several foreign institutions in connection with conferences. Dr. Aasma is also interested in approximation theory and dynamical systems in economics. He is a reviewer for several journals and databases of mathematics. He is a member of some mathematical societies, such as the Estonian Mathematical Society and the Estonian Operational Research Society. He teaches real analysis, complex analysis, operations research, mathematical economics, and financial mathematics. Dr. Aasma is the author of several textbooks for Estonian universities.
Hemen Dutta is a senior assistant professor of mathematics at Gauhati University, India. Dr. Dutta received his MSc and PhD in mathematics from Gauhati University, India. He received his MPhil in mathematics from Madurai Kamaraj University, India. Dr. Dutta's research interests include summability theory and functional analysis. He has to his credit several papers in research journals and two books. He visited foreign institutions in connection with research collaboration and conference. He has delivered talks at foreign and national institutions. He is a member on the editorial board of several journals and he is continuously reviewing for some databases and journals of mathematics. Dr. Dutta is a member of some mathematical societies.
P.N. Natarajan, Dr Radhakrishnan Awardee for the Best Teacher in Mathematics for the year 1990–91 by the Government of Tamil Nadu, India, has been working as an independent researcher after his retirement, in 2004, as professor and head, department of mathematics, Ramakrishna Mission Vivekananda College, Chennai, Tamil Nadu, India. Dr. Natarajan received his PhD in analysis from the University of Madras in 1980. He has to his credit over 100 research papers published in several reputed international journals. He authored a book (two editions) and contributed in an edited book. Dr. Natarajan's research interests include summability theory and functional analysis (both classical and ultrametric). Besides visiting several institutes of repute in Canada, France, Holland, and Greece on invitation, he has participated in several international conferences and has chaired sessions.
About the Book
This book is designed as a textbook for graduate students and researchers as a first course in summability theory. The book starts with a short and compact overview of basic results on summability theory and special summability methods. Then, results on matrix transforms of several matrix methods are discussed, which have not been widely discussed in textbooks yet. One of the most important applications of summability theory is the estimation of the speed of convergence of a sequence or series. In the textbooks published in English language until now, no description of the notions of convergence, boundedness, and summability with speed can be found, started by G. Kangro in 1969. Finally, this book discusses these concepts and some applications of these concepts in approximation theory. Each chapter of the book contains several solved examples and chapter-end exercises including hints for solution.
Chapter 1Introduction and General Matrix Methods
1.1 Brief Introduction
The study of the convergence of infinite series is an ancient art. In ancient times, people were more concerned with orthodox examinations of convergence of infinite series. Series that did not converge were of no interest to them until the advent of L. Euler (1707–1783), who took up a serious study of “divergent series”; that is, series that did not converge. Euler was followed by a galaxy of great mathematicians, such as C.F. Gauss (1777–1855), A.L. Cauchy (1789–1857), and N.H. Abel (1802–1829). The interest in the study of divergent series temporarily declined in the second half of the nineteenth century. It was rekindled at a later date by E. Cesàro, who introduced the idea of convergence in 1890. Since then, many other mathematicians have been contributing to the study of divergent series. Divergent series have been the motivating factor for the introduction of summability theory.
Summability theory has many uses in analysis and applied mathematics. An engineer or physicist who works with Fourier series, Fourier transforms, or analytic continuation can find summability theory very useful for his/her research.
Throughout this chapter, we assume that all indices and summation indices run from 0 to , unless otherwise specified. We denote sequences by {xk} or (xk), depending on convenience.
Consider the sequence
which is known to diverge. However, let
proving that
In this case, we say that the sequence converges to in the sense of Cesàro or is summable to . Similarly, consider the infinite series
The associated sequence of partial sums is , which is -summable to . In this case, we say that the series is -summable to .
With this brief introduction, we recall the following concepts and results.
1.2 General Matrix Methods
Definition 1.1
Given an infinite matrix , and a sequence , by the -transform of , we mean the sequence
where we suppose that the series on the right converges. If , we say that the sequence is summable or -summable to . If whenever , then is said to be preserving convergence for convergent sequences, or sequence-to-sequence conservative (for brevity, Sq-Sq conservative). If is sequence-to-sequence conservative with , we say that is sequence-to-sequence regular (shortly, Sq-Sq regular). If , whenever, , then is said to preserve the convergence of series, or series-to-sequence conservative (i.e., Sr-Sq conservative). If is series-to-sequence conservative with , we say that is series-to-sequence regular (shortly, Sr-Sq regular).
In this chapter and in Chapters 2 and 3, for conservative and regular, we mean only Sq–Sq conservativity and Sq-Sq regularity.
If are sequence spaces, we write
if is defined and , whenever, . With this notation, if is conservative, we can write , where denotes the set of all convergent sequences. If is regular, we write
denoting the “preservation of limit.”
Definition 1.2
A method is said to be lower triangular (or simply, triangular) if for , and normal if is lower triangular if for every .
Example 1.1
Let be the Zweier method; that is, , defined by the lower triangular method where (see [2], p. 14) and
for . The method is regular. The transformation for can be presented as
Then,
for every that is, .
We now prove a landmark theorem in summability theory due to Silverman–Toeplitz, which characterizes a regular matrix in terms of the entries of the matrix (see [3–5]).
Theorem 1.1 (Silverman-Toeplitz)
is regular, that is, , if and only if
and
with and .
Proof
Sufficiency. Assume that conditions (1.1)–(1.3) with and hold. Let with . Since converges, it is bounded; that is, , , or, equivalently, , for all .
Now
in view of (1.1), and so
is defined. Now
Since , given an , there exists an , where denotes the set of all positive integers, such that
where is such that
and hence
Using (1.5) and (1.6), we obtain
By (1.2), there exists a positive integer such that
This implies that
Consequently, for every , we have
Thus,
Taking the limit as in (1.4), we have, by (1.7), that
since . Hence, is regular, completing the proof of the sufficiency part.
Necessity. Let be regular. For every fixed , consider the sequence , where
For this sequence , . Since and is regular, it follows that . Again consider the sequence , where for all . Note that . For this sequence , . Since and is regular, we have . It remains to prove (1.1). First, we prove that converges. Suppose not. Then, there exists an such that
In fact, diverges to . So we can find a strictly increasing sequence of positive integers such that
Define the sequence by
Note that and converges. In particular, converges. However,
This leads to a contradiction since diverges. Thus,
To prove that (1.1) holds, we assume that
and arrive at a contradiction.
We construct two strictly increasing sequences and of positive integers in the following manner.
Let . Since , choose such that
Having chosen the positive integers and , , , choose positive integers and such that
and
