Summability theory and its applications

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Foreword
Preface to Second Edition
List of Tables
List of Abbreviations and Symbols
Author
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 !
		2.2.2. The Space `1
		2.2.3. The Spaces f and f0
		2.2.4. The Spaces c and c0
		2.2.5. The Space `p
		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 !p 0, !p and !p 1
	2.3. NORMED SEQUENCE SPACES
	2.4. PARANORMED SEQUENCE SPACES
		2.4.1. The Spaces `1(p), c(p) and c0(p)
		2.4.2. The Space `(p)
		2.4.3. The Spaces !1(p), !(p) and !0(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. Cesaro Means
		3.5.3. Euler Means
		3.5.4. Taylor Matrices
		3.5.5. Riesz Means
		3.5.6. Norlund 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. CESARO SEQUENCE SPACES AND CONCERNING DUALITY RELATION
		4.2.1. The Cesaro Sequence Spaces of Non-absolute Type
		4.2.2. The -, - and 
-Duals of the Spaces ec0 and e
		4.2.3. The Characterization of Some Matrix Mappings Related to the Space ec
	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 b`1, bc, bc0 and b`1
		4.4.3. Domain of Generalized Difference Matrix B(r, s) in the Spaces f0 and f
		4.4.4. The Sequence Spaces b f0 and b 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 `1 (
m), c (
m) and c0 (
m)
		4.5.2. Matrix Transformations
		4.5.3. 
m-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 `p ????
(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 ar p, ar0, arc and ar 1 of Non-absolute Type
		4.6.2. The Inclusion Relations
		4.6.3. The -, - and -Duals of the Spaces ar p, ar0, arc and ar 1
		4.6.4. Some Matrix Mappings on the Spaces ar p and arc
	4.7. RIESZ SEQUENCE SPACES AND CONCERNING DUALITY RELATION
		4.7.1. The Riesz Sequence Spaces rt(p), rt0 (p), rtc (p) and rt1(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 er
	4.9. DOMAIN OF THE GENERALIZED WEIGHTED MEAN AND CONCERNING DUALITY RELATION
		4.9.1. Some Matrix Transformations Related to the Sequence Spaces (u, v; p)
	4.10. DOMAINS OF TRIANGLES IN THE SPACES OF STRONGLY C1–SUMMABLE ...
		4.10.1. Matrix Transformations on wp 0, wp and wp
		4.10.2. The -Duals of wp 0(U), wp(U) and wp1 (U)
		4.10.3. Matrix Transformations on the Spaces wp 0(U), wp(U) and wp1(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
		5.2.3. Goldberg’s Classification of Spectrum
	5.3. THE FINE SPECTRUM OF THE CESARO OPERATOR IN THE SPACES c0 AND c
	5.4. THE FINE SPECTRA OF THE DIFFERENCE OPERATOR 
(1) ON THE SPACE `p
	5.5. THE FINE SPECTRA OF THE DIFFERENCE OPERATOR 
(1) ON THE SPACE bvp
	5.6. THE FINE SPECTRA OF THE CESARO 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 `p AND bvp
		5.8.1. The Fine Spectrum of the Operator B(r, s, t) on the Sequence Space `p, (1 < p < 1).
		5.8.2. The Spectrum of the Operator B(r, s, t) on the Sequence Space bvp, (1 < p < 1)
	5.9. CONCLUSION
CHAPTER 6: Core of a Sequence
	6.1. KNOPP CORE
	6.2. ˙-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 - AND -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 -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
		9.4.1. An Application of Quasi Power Increasing Sequences
Bibliography
Index