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E-grāmata: Summability Theory and Its Applications

(Professor, Inonu University, Turkey)
  • Formāts: 520 pages
  • Izdošanas datums: 27-Jun-2022
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781000599183
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Summability Theory and Its ApplicationsPalielināt
  • Formāts: 520 pages
  • Izdošanas datums: 27-Jun-2022
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781000599183
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"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. Thissubstantially 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.



This book 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.

 

 

Foreword xv
Preface to Second Edition xvii
List of Tables
xxi
List of Abbreviations and Symbols
xxiii
Author xxvii
Chapter 1 Infinite Matrices
1(14)
1.1 Preliminaries
1(2)
1.1.1 Some Problems Involving the Use of Infinite Matrices
2(1)
1.2 Some Definitions
3(1)
1.3 Some Characteristic Properties Of Infinite Matrices
4(2)
1.4 Some Special Infinite Matrices
6(1)
1.5 The Structure Of An Infinite Matrix
7(1)
1.6 The Exponential Function Of A Lower-Semi Matrix
8(1)
1.7 Semi-Continuous And Continuous Matrices
8(1)
1.8 Inverses Of Infinite Matrices
9(6)
1.8.1 Inverses of Lower Semi-Matrices
9(6)
Chapter 2 Normed and Paranormed Sequence Spaces
15(22)
2.1 Linear Sequence Spaces
15(1)
2.2 Metric Sequence Spaces
16(3)
2.2.1 The Space ω
16(1)
2.2.2 The Space ∞
16(1)
2.2.3 The Spaces ƒ and ƒ0
17(1)
2.2.4 The Spaces c and Co
17(1)
2.2.5 The Space lp
17(1)
2.2.6 The Space bs
18(1)
2.2.7 The Spaces cs and cs0
18(1)
2.2.8 The Space bυ1
18(1)
2.2.9 The Spaces ωp0, ωp and ωp∞
19(1)
2.3 Normed Sequence Spaces
19(3)
2.4 Paranormed Sequence Spaces
22(3)
2.4.1 The Spaces l∞(p), c(p) and c0(p)
23(1)
2.4.2 The Space l(p)
23(1)
2.4.3 The Spaces W∞(p), ω{p) and ω0(p)
24(1)
2.4.4 The Spaces bs(p), cs(p) and cs0{p)
24(1)
2.5 The Dual Spaces of a Sequence Space
25(12)
Chapter 3 Matrix Transformations in Sequence Spaces
37(22)
3.1 Introduction
37(1)
3.2 Introduction to Summability
38(3)
3.2.1 Summability
38(3)
3.3 Characterizations of Some Matrix Classes
41(8)
3.4 Dual Summability Methods
49(4)
3.4.1 Dual Summability Methods Dependent on a Stieltjes Integral
49(1)
3.4.2 Relation Between the Dual Summability Methods
50(1)
3.4.3 Usual Dual Summability Methods
51(2)
3.5 Some Examples of Toeplitz Matrices
53(6)
3.5.1 Arithmetic Means
53(1)
3.5.2 Cesaro Means
53(1)
3.5.3 Euler Means
54(1)
3.5.4 Taylor Matrices
55(1)
3.5.5 Riesz Means
55(1)
3.5.6 Norlund Means
56(1)
3.5.7 Ar Matrices
56(1)
3.5.8 Hausdorff Matrices
56(1)
3.5.9 Borel Matrix
57(1)
3.5.10 Abel Matrix (cf. Peyerimhoff [ 317, p. 24])
57(2)
Chapter 4 Matrix Domains in Sequence Spaces
59(162)
4.1 Preliminaries, Background and Notations
59(4)
4.2 Cesaro Sequence Spaces and Concerning Duality Relation
63(6)
4.2.1 The Cesaro Sequence Spaces of Non-absolute Type
64(2)
4.2.2 The α-, β- and γ-Duals of the Spaces c0 and c
66(2)
4.2.3 The Characterization of Some Matrix Mappings Related to the Space c
68(1)
4.3 Difference Sequence Spaces and Concerning Duality Relation
69(13)
4.3.1 The Space bυp of Sequences of p-Bounded Variation
71(3)
4.3.2 The Dual Spaces of the Space bυp
74(6)
4.3.3 Certain Matrix Mappings Related to the Sequence Space bυp
80(2)
4.4 Domain of Generalized Difference Matrix B(R, S)
82(16)
4.4.1 Domain of Generalized Difference Matrix B(r, s) in the Classical Sequence Spaces
83(4)
4.4.2 Some Matrix Transformations Related to the Sequence Spaces l∞, c;, c0 and l1
87(2)
4.4.3 Domain of Generalized Difference Matrix B(r, s) in the Spaces ƒ0 and ƒ
89(2)
4.4.4 The Sequence Spaces ƒo and ƒ Derived by the Domain of the Matrix B(r, s)
91(3)
4.4.5 Some Matrix Transformations Related to the Sequence Space ƒ
94(4)
4.5 Spaces Of Difference Sequences Of Order m
98(38)
4.5.1 Dual Spaces of l∞ (Δm), c(Δm) and c0(Δm)
103(7)
4.5.2 Matrix Transformations
110(3)
4.5.3 Δm-Statistical Convergence
113(5)
4.5.4 Paranormed Difference Sequence Spaces
118(5)
4.5.5 The Space of p-Summable Difference Sequences of Order m
123(5)
4.5.6 Certain Matrix Mappings on the Sequence Space lP(Δ(m))
128(2)
4.5.7 υ-Invariant Sequence Spaces
130(2)
4.5.8 Paranormed Difference Sequence Spaces Generated by Moduli and Orlicz Functions
132(4)
4.6 The Domain of the Matrix Ar and Concerning Duality Relation
136(12)
4.6.1 The Sequence Spaces arp, ar0, arc and ar∞ of Non-absolute Type
137(1)
4.6.2 The Inclusion Relations
138(2)
4.6.3 The α, β- and γ-Duals of the Spaces arp, ar0, arc and ar∞
140(3)
4.6.4 Some Matrix Mappings on the Spaces arp and arc
143(5)
4.7 Riesz Sequence Spaces and Concerning Duality Relation
148(17)
4.7.1 The Riesz Sequence Spaces rt(p), rt0(p), rtc(p) and rt∞(p) of Non-absolute Type
148(7)
4.7.2 Matrix Mappings Related to the Riesz Sequence Spaces
155(10)
4.8 Euler Sequence Spaces and Concerning Duality Relation
165(18)
4.8.1 Euler Sequence Spaces of Non-absolute Type
165(8)
4.8.2 Certain Matrix Transformations Related to the Euler Sequence Spaces
173(7)
4.8.3 Some Geometric Properties of the Space erp
180(3)
4.9 Domain of the Generalized Weighted Mean and Concerning Duality Relation
183(12)
4.9.1 Some Matrix Transformations Related to the Sequence Spaces λ(u, υ;p)
188(7)
4.10 Domains of Triangles in the Spaces of Strongly C1-Summable ...
195(16)
4.10.1 Matrix Transformations on Wpq, Wp and Wp∞
198(6)
4.10.2 The β-Duals of Wp0(U), Wp(U) and Wp∞(U)
204(4)
4.10.3 Matrix Transformations on the Spaces Wpq(U), Wp(U) and Wp∞(U)
208(2)
4.10.4 Conclusion
210(1)
4.11 Characterizations of Some Other Classes of Matrix Transformations
211(7)
4.12 Conclusion
218(3)
Chapter 5 Spectrum of Some Particular Matrices
221(44)
5.1 Preliminaries, Background and Notations
221(1)
5.2 Subdivisions of the Spectrum
222(4)
5.2.1 The Point Spectrum, Continuous Spectrum and Residual Spectrum
222(1)
5.2.2 The Approximate Point Spectrum, Defect Spectrum and Compression Spectrum
222(2)
5.2.3 Goldberg's Classification of Spectrum
224(2)
5.3 The Fine Spectrum of the Cesaro Operator in the Spaces c0 and c
226(5)
5.4 The Fine Spectra of the Difference Operator Δ(1) On the Space lp
231(3)
5.5 The Fine Spectra of the Difference Operator Δ(1) on the Space bυp
234(4)
5.6 The Fine Spectra of the Cesaro Operator C1 on the Space bυp
238(5)
5.7 The Spectrum of the Operator B(r, s) on the Spaces c0 and c
243(9)
5.7.1 The Generalized Difference Operator B(r, s)
244(8)
5.8 The Fine Spectra of the Operator B(r, s, t) on the Spaces lp AND bυp
252(12)
5.8.1 The Fine Spectrum of the Operator B(r, s, t) on the Sequence Space lp, (1 > p > ∞)
252(9)
5.8.2 The Spectrum of the Operator B(r, s, t) on the Sequence Space bυp, (1 > p > ∞)
261(3)
5.9 Conclusion
264(1)
Chapter 6 Core of a Sequence
265(52)
6.1 Knopp Core
265(36)
6.2 σ-Core
301(3)
6.3 I-Core
304(7)
6.4 FB-Core
311(6)
Chapter 7 Double Sequences
317(52)
7.1 Preliminaries, Background and Notations
317(3)
7.2 Pringsheim Convergence of Double Series
320(16)
7.2.1 Absolute Convergence
321(7)
7.2.2 Cauchy Product
328(8)
7.3 The Double Sequence Space Lq
336(5)
7.4 Some New Spaces of Double Sequences
341(7)
7.5 The Spaces CSP, CSbp, CSr and BV of Double Series
348(3)
7.6 The α- and β-Duals of the Spaces of Double Series
351(4)
7.7 Characterization of Some Classes of Four-Dimensional Matrices
355(2)
7.8 Binomial Spaces of Double Sequences
357(10)
7.8.1 Dual Spaces of the Binomial Spaces
360(5)
7.8.2 Characterizations of Some Matrix Classes
365(2)
7.9 Conclusion
367(2)
Chapter 8 Sequences of Fuzzy Numbers
369(72)
8.1 Introduction
369(1)
8.2 Convergence of a Sequence of Fuzzy Numbers
370(16)
8.2.1 The Limit Superior and Limit Inferior of a Sequence of Fuzzy Numbers
378(4)
8.2.2 The Core of a Sequence of Fuzzy Numbers
382(4)
8.3 Statistical Convergence of a Sequence of Fuzzy Numbers
386(18)
8.3.1 Statistical Convergence of a Sequence of Fuzzy Numbers and the Statistical Convergence of the Corresponding Sequence of α-Cuts
387(1)
8.3.2 Statistically Monotonic and Statistically Bounded Sequences of Fuzzy Numbers
388(5)
8.3.3 Statistical Cluster Points and Statistical Limit Points of a Sequence of Fuzzy Numbers
393(2)
8.3.4 The Statistical Limit Inferior and the Statistical Limit Superior of a Statistically Bounded Sequence of Fuzzy Numbers
395(5)
8.3.5 Further Results
400(1)
8.3.6 Relation Between Statistical Cluster Points and Statistical Extreme Limit Points
401(3)
8.4 The Classical Sets of Sequences of Fuzzy Numbers
404(24)
8.4.1 Preliminaries, Background and Notations
406(5)
8.4.2 Determination of Duals of the Classical Sets of Sequences of Fuzzy Numbers
411(6)
8.4.3 Matrix Transformations Between Some Sets of Sequences of Fuzzy Numbers
417(11)
8.5 Quasilinearity of the Classical Sets of Sequences of Fuzzy Numbers
428(6)
8.5.1 The Quasilinearity of the Classical Sets of Sequences of Fuzzy Numbers
431(3)
8.6 Certain Sets of Sequences of Fuzzy Numbers Defined by a Modulus
434(5)
8.6.1 The Spaces of Sequences of Fuzzy Numbers Defined by a Modulus Function
435(4)
8.7 Conclusion
439(2)
Chapter 9 Absolute Summability
441(22)
9.1 Background, Preliminaries and Notations
441(1)
9.2 Absolute Summability of Sequences and Series
442(5)
9.3 Inclusion Theorems
447(4)
9.4 Summability Factors Theorems
451(12)
9.4.1 An Application of Quasi Power Increasing Sequences
455(8)
Bibliography 463(26)
Index 489
Dr. Feyzi Baar is a Professor Emeritus since July 2016 at nönü University, Turkey. He has published three books for graduate students and researchers and more than 160 scientific papers in the field of summability theory, sequence spaces, FK-spaces, Schauder bases, dual spaces, matrix transformations, spectrum of certain linear operators represented by a triangle matrix over some sequence spaces, the alpha-, beta- and gamma-duals and some topological properties of the domains of some double and four-dimensional triangles in certain spaces of single and double sequences and sets of the sequences of fuzzy numbers. Nowadays, Professor Baar works on the development of sequences and series, and the basic concepts of summability in non-newtonian calculus. He has guided 17 MA and 10 Ph.D. students and served as a referee for 141 international scientific journals. He is reviewer Mathematical Reviews since 2007 and Zentralblatt MATH, and the member of editorial boards of 21 scientific journals. He is also a member of scientific committees of 17 mathematics conferences, delivered talks at 14 different universities as an invited speaker, and worked on 10 scientific project, and participated in more than 70 mathematics symposiums with papers.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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