Classical Summability Theory

139 Pages·2017·0.733 MB·English

by  P.N. Natarajan

P.N. Natarajan
Classical
Summability
Theory
Classical Summability Theory
P.N. Natarajan
Classical Summability
Theory
123
P.N. Natarajan
FormerlyoftheDepartmentofMathematics
Ramakrishna MissionVivekanandaCollege
Chennai, Tamil Nadu
India
ISBN978-981-10-4204-1 ISBN978-981-10-4205-8 (eBook)
DOI 10.1007/978-981-10-4205-8
LibraryofCongressControlNumber:2017935546
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Dedicated to my wife Vijayalakshmi and my
children Sowmya and Balasubramanian
Preface
Thestudyofconvergenceofinfiniteseriesisaveryoldart.Inancienttimes,people
were interested in orthodox examination for convergence of infinite series.
Divergent series, i.e., infinite series which do not converge, was of no interest to
them until the advent of L. Euler (1707–1783), who took up a serious study of
divergent series. He was later followed by a galaxy of very great mathematicians.
Study of divergent series is the foundation of summability theory. Summability
theory has many utilities in analysis and applied mathematics. An engineer or
physicist,whoworksonFourierseries,Fouriertransformsoranalyticcontinuation,
can find summability theory very useful for his/her research.
In the present book, some of the contributions of the author to classical
summability theory are highlighted, thereby supplementing, the material already
available in standard texts on summability theory.
Therearesixchaptersinall.Thesalientfeaturesofeachchapterarelistedbelow.
In Chap. 1, after a very brief introduction, we recall well-known definitions and
concepts. We state and prove Silverman–Toeplitz theorem, Schur’s theorem and
then deduce Steinhaus theorem. We introduce a sequence space Λ , r(cid:1)1 being a
r
fixedintegerandmakeadetailedstudyofthespaceΛ ,especiallyfromthepointof
r
view of sequences of zeros and ones. We prove a Steinhaus type result involving
the space Λ , which improves Steinhaus theorem. We prove some more Steinhaus
r
type theorems too.
Chapter 2 deals with the core of a sequence. We present an improvement of
Sherbakhoff’sresult,whichleadstoashortandveryelegantproofofKnopp’score
theorem.Wealsopresentsomenicepropertiesoftheclassð‘;‘Þofinfinitematrices.
Chapter3isdevotedtoadetailedstudyofsomespecialmethodsofsummability,
viz., the Abel method, the Weighted mean method, the Euler method and the
ðM;‚ Þ or Natarajan method. We bring out the connection between the Abel
n
method and the Natarajan method. Some product theorems involving certain
summability methods are also proved.
InChap.4,somenicerpropertiesoftheðM;‚ Þmethodareestablished.Further,
n
we prove a few results on the Cauchy multiplication of certain summable series.
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viii Preface
InChap. 5,a new definition of convergenceofa double sequence and adouble
series is introduced. In the context of this new definition, Silverman–Toeplitz
theorem for 4-dimensional infinite matrices is proved. We also prove Schur’s and
Steinhaus theorems for 4-dimensional infinite matrices.
Finally in Chap. 6, we introduce the Nörlund, the Weighted mean and the
ðM;‚m;nÞ or Natarajan methods for double sequences and double series and study
some of their properties.
I thank my mentor Prof. M.S. Rangachari for initiating me to the topic of
summability theory—both classical and ultrametric. I thank Mr. E. Boopal for
typing the manuscript.
Chennai, India P.N. Natarajan
Contents
1 General Summability Theory and Steinhaus Type Theorems .. .... 1
1.1 Basic Definitions and Concepts .. .... .... .... .... ..... .... 1
1.2 The Silverman–Toeplitz Theorem, Schur’s Theorem,
and Steinhaus Theorem..... .... .... .... .... .... ..... .... 3
1.3 A Steinhaus Type Theorem . .... .... .... .... .... ..... .... 13
1.4 The Role Played by the Sequence Spaces Λr.... .... ..... .... 16
1.5 More Steinhaus Type Theorems.. .... .... .... .... ..... .... 19
References.. .... .... .... ..... .... .... .... .... .... ..... .... 25
2 Core of a Sequence and the Matrix Class ð‘;‘Þ .... .... ..... .... 27
2.1 Core of a Sequence .. ..... .... .... .... .... .... ..... .... 27
2.2 Natarajan’s Theorem and Knopp’s Core Theorem .... ..... .... 28
2.3 Some Results for the Matrix Class ð‘;‘Þ ... .... .... ..... .... 31
2.4 A Mercerian Theorem. ..... .... .... .... .... .... ..... .... 35
References.. .... .... .... ..... .... .... .... .... .... ..... .... 36
3 Special Summability Methods... .... .... .... .... .... ..... .... 37
3.1 Weighted Mean Method.... .... .... .... .... .... ..... .... 37
3.2 ðM;‚ Þ Method or Natarajan Method.. .... .... .... ..... .... 48
n
3.3 The Abel Method and the ðM;‚nÞ Method.. .... .... ..... .... 54
3.4 The Euler Method and the ðM;‚nÞ Method . .... .... ..... .... 55
References.. .... .... .... ..... .... .... .... .... .... ..... .... 61
4 MorePropertiesoftheðM;‚nÞMethodandCauchyMultiplication
of Certain Summable Series.... .... .... .... .... .... ..... .... 63
4.1 Some Nice Properties of the ðM;‚ Þ Method.... .... ..... .... 63
n
4.2 Iteration of ðM;‚ Þ Methods .... .... .... .... .... ..... .... 69
n
4.3 Cauchy Multiplication of ðM;‚ Þ-Summable Series... ..... .... 73
n
4.4 Cauchy Multiplication of Euler Summable Series. .... ..... .... 77
References.. .... .... .... ..... .... .... .... .... .... ..... .... 82
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x Contents
5 The Silverman–Toeplitz, Schur, and Steinhaus Theorems
for Four-Dimensional Infinite Matrices... .... .... .... ..... .... 83
5.1 A New Definition of Convergence of a Double Sequence
and a Double Series.. ..... .... .... .... .... .... ..... .... 83
5.2 The Silverman–Toeplitz Theorem for Four-Dimensional Infinite
Matrices ... .... .... ..... .... .... .... .... .... ..... .... 85
5.3 TheSchurandSteinhausTheoremsforFour-DimensionalInfinite
Matrices ... .... .... ..... .... .... .... .... .... ..... .... 94
References.. .... .... .... ..... .... .... .... .... .... ..... .... 100
6 The Nörlund, Weighted Mean, and ðM;‚m;nÞ Methods
for Double Sequences .... ..... .... .... .... .... .... ..... .... 101
6.1 Nörlund Method for Double Sequences .... .... .... ..... .... 101
6.2 Weighted Mean Method for Double Sequences .. .... ..... .... 107
6.3 ðM;‚m;nÞ or Natarajan Method for Double Sequences . ..... .... 120
References.. .... .... .... ..... .... .... .... .... .... ..... .... 128
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 129
About the Author
P.N. Natarajan formerly with the Department of Mathematics, Ramakrishna
MissionVivekanandaCollege,Chennai,India,hasbeenanindependentresearcher
andmathematiciansincehisretirementin2004.HedidhisPh.D.attheUniversity
of Madras, under Prof. M.S. Rangachari, former director and head of the
RamanujanInstituteforAdvancedStudyinMathematics,UniversityofMadras.An
activeresearcher,Prof.Natarajanhaspublishedover100researchpapersinseveral
international journals like the Proceedings of the American Mathematical Society,
BulletinoftheLondonMathematicalSociety,IndagationesMathematicae,Annales
Mathematiques Blaise Pascal and Commentationes Mathematicae (Prace
Matematyczne). His research interests include summability theory and functional
analysis(both classical andultrametric).ProfessorNatarajanwashonored withthe
Dr. RadhakrishnanAward fortheBest TeacherinMathematics fortheyear1990–
1991bytheGovernmentofTamilNadu.Inadditiontobeinginvitedtovisitseveral
renowned institutes in Canada, France, Holland and Greece, Prof. Natarajan has
participated in several international conferences and chaired sessions. He has
authored two books, An Introduction to Ultrametric Summability Theory and its
second edition, both published with Springer in 2013 and 2015, respectively.
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