(Ebook) Summability theory and its applications 2nd Edition by Feyzi Basar ISBN 1003294154 9781003294153

(Ebook) Summability theory and its applications 2nd Edition by Feyzi Basar ISBN 1003294154 9781003294153

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ISBN 10: 1003294154 
ISBN 13: 9781003294153
Author: Feyzi Basar

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.

(Ebook) Summability theory and its applications 2nd Table of contents:

Chapter 1  Infinite Matrices
1.1 Preliminaries
1.1.1 Some Problems Involving the Use of Infinite Matrices
1.2 Some Definitions
1.3 Some Characteristic Properties Of Infinite Matrices
1.4 Some Special Infinite Matrices
1.5 The Structure Of An Infinite Matrix
1.6 The Exponential Function Of A Lower-Semi Matrix
1.7 Semi-Continuous And Continuous Matrices
1.8 Inverses Of Infinite Matrices
1.8.1 Inverses of Lower Semi-Matrices
Chapter 2  Normed and Paranormed Sequence Spaces
2.1 Linear Sequence Spaces
2.2 Metric Sequence Spaces
2.2.1 The Space ω
2.2.2 The Space ℓ∞
2.2.3 The Spaces f and f0
2.2.4 The Spaces c and c0
2.2.5 The Space ℓp
2.2.6 The Space bs
2.2.7 The Spaces cs and cs0
2.2.8 The Space bv1
2.2.9 The Spaces ω0p, ωp and ω∞p
2.3 Normed Sequence Spaces
2.4 Paranormed Sequence Spaces
2.4.1 The Spaces ℓ∞(p), c(p) and c0(p)
2.4.2 The Space ℓ(p)
2.4.3 The Spaces ω∞(p), ω(p)$and$ω0(p)
2.4.4 The Spaces bs(p), cs(p) and cs0(p)
2.5 The Dual Spaces Of A Sequence Space
Chapter 3 Matrix Transformations in Sequence Spaces
3.1 Introduction
3.2 Introduction To Summability
3.2.1 Summability
3.3 Characterizations Of Some Matrix Classes
3.4 Dual Summability Methods
3.4.1 Dual Summability Methods Dependent on a Stieltjes Integral
3.4.2 Relation Between the Dual Summability Methods
3.4.3 Usual Dual Summability Methods
3.5 Some Examples Of Toeplitz Matrices
3.5.1 Arithmetic Means
3.5.2 Cesàro Means
3.5.3 Euler Means
3.5.4 Taylor Matrices
3.5.5 Riesz Means
3.5.6 Nörlund Means
3.5.7 Ar Matrices
3.5.8 Hausdorff Matrices
3.5.9 Borel Matrix
3.5.10 Abel Matrix (cf. Peyerimhoff [317, p. 24])
Chapter 4  Matrix Domains in Sequence Spaces
4.1 Preliminaries, Background And Notations
4.2 Cesàro Sequence Spaces And Concerning Duality Relation
4.2.1 The Cesàro Sequence Spaces of Non-absolute Type
4.2.2 The α-, β- and γ-Duals of the Spaces c˜0 and c˜
4.2.3 The Characterization of Some Matrix Mappings Related to the Space c˜
4.3 Difference Sequence Spaces And Concerning Duality Relation
4.3.1 The Space bvp of Sequences of p-Bounded Variation
4.3.2 The Dual Spaces of the Space bvp
4.3.3 Certain Matrix Mappings Related to the Sequence Space bvp
4.4 Domain Of Generalized Difference Matrix B(r,s)
4.4.1 Domain of Generalized Difference Matrix B(r,s) in the Classical Sequence Spaces
4.4.2 Some Matrix Transformations Related to the Sequence Spaces ℓ^∞, c^, c^0 and ℓ^1
4.4.3 Domain of Generalized Difference Matrix B(r,s) in the Spaces f0 and f
4.4.4 The Sequence Spaces f^0 and f^ Derived by the Domain of the Matrix B(r,s)
4.4.5 Some Matrix Transformations Related to the Sequence Space f^
4.5 Spaces Of Difference Sequences Of Order m
4.5.1 Dual Spaces of ℓ∞Δm, cΔm and c0Δm
4.5.2 Matrix Transformations
4.5.3 Δm-Statistical Convergence
4.5.4 Paranormed Difference Sequence Spaces
4.5.5 The Space of p-Summable Difference Sequences of Order m
4.5.6 Certain Matrix Mappings on the Sequence Space ℓp(Δ(m))
4.5.7 v-Invariant Sequence Spaces
4.5.8 Paranormed Difference Sequence Spaces Generated by Moduli and Orlicz Functions
4.6 The Domain Of The Matrix Ar And Concerning Duality Relation
4.6.1 The Sequence Spaces apr, a0r, acr and a∞r of Non-absolute Type
4.6.2 The Inclusion Relations
4.6.3 The α-, β- and γ-Duals of the Spaces apr, a0r, acr and a∞r
4.6.4 Some Matrix Mappings on the Spaces apr and acr
4.7 Riesz Sequence Spaces And Concerning Duality Relation
4.7.1 The Riesz Sequence Spaces rt(p), r0t(p), rct(p) and r∞t(p) of Non--absolute Type
4.7.2 Matrix Mappings Related to the Riesz Sequence Spaces
4.8 Euler Sequence Spaces And Concerning Duality Relation
4.8.1 Euler Sequence Spaces of Non-absolute Type
4.8.2 Certain Matrix Transformations Related to the Euler Sequence Spaces
4.8.3 Some Geometric Properties of the Space epr
4.9 Domain Of The Generalized Weighted Mean And Concerning Duality Relation
4.9.1 Some Matrix Transformations Related to the Sequence Spaces λ(u,v;p)
4.10 Domains Of Triangles In The Spaces Of Strongly C1--Summable ...
4.10.1 Matrix Transformations on w0p, wp and w∞p
4.10.2 The β-Duals of w0p(U), wp(U) and w∞p(U)
4.10.3 Matrix Transformations on the Spaces w0p(U), wp(U) and w∞p(U)
4.10.4 Conclusion
4.11 Characterizations Of Some Other Classes Of Matrix Transformations
4.12 Conclusion
Chapter 5 Spectrum of Some Particular Matrices
5.1 Preliminaries, Background And Notations
5.2 Subdivisions Of The Spectrum
5.2.1 The Point Spectrum, Continuous Spectrum and Residual Spectrum
5.2.2 The Approximate Point Spectrum, Defect Spectrum and Compression Spectrum
5.2.3 Goldberg's Classification of Spectrum
5.3 The Fine Spectrum Of The Cesàro Operator In The Spaces c0 And c
5.4 The Fine Spectra Of The Difference Operator Δ(1) On The Space ℓp
5.5 The Fine Spectra Of The Difference Operator Δ(1) On The Space bvp
5.6 The Fine Spectra Of The Cesàro Operator C1 On The Space bvp
5.7 The Spectrum Of The Operator B(r,s) On The Spaces c0 And~c
5.7.1 The Generalized Difference Operator B(r,s)
5.8 The Fine Spectra Of The Operator B(r,s,t) On The Spaces ℓp and bvp
5.8.1 The Fine Spectrum of the Operator B(r,s,t) on the Sequence Space ℓp, (15.8.2 The Spectrum of the Operator B(r,s,t) on the Sequence Space bvp, (15.9 Conclusion
Chapter 6  Core of a Sequence
6.1 Knopp Core
6.2 σ-Core
6.3 I-Core
6.4 FB-Core
Chapter 7  Double Sequences
7.1 Preliminaries, Background And Notations
7.2 Pringsheim Convergence Of Double Series
7.2.1 Absolute Convergence
7.2.2 Cauchy Product
7.3 The Double Sequence Space Lq
7.4 Some New Spaces Of Double Sequences
7.5 The Spaces CSp, CSbp, CSr and BV Of Double Series
7.6 The α- And β-Duals Of The Spaces Of Double Series
7.7 Characterization Of Some Classes Of Four-Dimensional Matrices
7.8 Binomial Spaces Of Double Sequences
7.8.1 Dual Spaces of the Binomial Spaces
7.8.2 Characterizations of Some Matrix Classes
7.9 Conclusion
Chapter 8 Sequences of Fuzzy Numbers
8.1 Introduction
8.2 Convergence Of A Sequence Of Fuzzy Numbers
8.2.1 The Limit Superior and Limit Inferior of a Sequence of Fuzzy Numbers
8.2.2 The Core of a Sequence of Fuzzy Numbers
8.3 Statistical Convergence Of A Sequence Of Fuzzy Numbers
8.3.1 Statistical Convergence of a Sequence of Fuzzy Numbers and the Statistical Convergence of the Corresponding Sequence of α-Cuts
8.3.2 Statistically Monotonic and Statistically Bounded Sequences of Fuzzy Numbers
8.3.3 Statistical Cluster Points and Statistical Limit Points of a Sequence of Fuzzy Numbers
8.3.4 The Statistical Limit Inferior and the Statistical Limit Superior of a Statistically Bounded Sequence of Fuzzy Numbers
8.3.5 Further Results
8.3.6 Relation Between Statistical Cluster Points and Statistical Extreme Limit Points
8.4 The Classical Sets Of Sequences Of Fuzzy Numbers
8.4.1 Preliminaries, Background and Notations
8.4.2 Determination of Duals of the Classical Sets of Sequences of Fuzzy Numbers
8.4.3 Matrix Transformations Between Some Sets of Sequences of Fuzzy Numbers
8.5 Quasilinearity Of The Classical Sets Of Sequences Of Fuzzy Numbers
8.5.1 The Quasilinearity of the Classical Sets of Sequences of Fuzzy Numbers
8.6 Certain Sets Of Sequences Of Fuzzy Numbers Defined By A Modulus
8.6.1 The Spaces of Sequences of Fuzzy Numbers Defined by a Modulus Function
8.7 Conclusion
Chapter 9 Absolute Summability
9.1 Background, Preliminaries And Notations
9.2 Absolute Summability Of Sequences And Series
9.3 Inclusion Theorems
9.4 Summability Factors Theorems

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