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Summability Theory and its Applications

  • ID: 2317294
  • Book
  • January 2012
  • Bentham Science Publishers Ltd
The theory of summability has many uses throughout analysis and applied mathematics. Engineers and physicists working with Fourier series or analytic continuation will also find the concepts of summability theory valuable to their research.
The concepts of summability have been extended to the sequences of fuzzy numbers and also to the theorems of ergodic theory. This e-book explains various aspects of summability and demonstrates applications in a coherent manner. The content can readily serve as a useful series of lecture notes on the subject.
This e-book comprises of 8 chapters starting from classical sequence spaces and covering matrix transformations and fuzzy numbers. An accompanying bibliography with extensive references makes this a valuable source of information for readers interested in summability theory as well as other branches of science.
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Foreword
Preface
Acknowledgements
Chapter 1. Background and Notations
1.1. In nite Matrices
1.2. Some De nitions
1.3. Some Characteristic Properties of In nite Matrices
1.4. Some Special In nite Matrices
1.5. The Structure of an In nite Matrix
1.6. The Exponential Function of a Lower-semi Matrix
1.7. Semi-continuous and Continuous Matrices
1.8. Inverses of In nite Matrices
Chapter 2. Normed and Paranormed Sequence Spaces
2.1. Linear Sequence Spaces
2.2. Metric Sequence Spaces
2.3. Normed Sequence Spaces
2.4. Paranormed Sequence Spaces
2.5. The Dual Spaces of a Sequence Space
Chapter 3. Matrix Transformations in Sequence Spaces
3.1. Introduction
3.2. Introduction to Summability
3.3. Characterizations of Some Matrix Classes
3.4. Dual Summability Methods
3.5. Some Examples of Toeplitz Matrices
Chapter 4. Matrix Domains in Sequence Spaces
4.1. Preliminaries, Background and Notations
4.2. Cesàro Sequence Spaces and Concerning Duality Relation
4.3. Di erence Sequence Spaces and Concerning Duality Relation
4.4. Domain of Generalized Di erence Matrix B(r; s)
4.5. Spaces of Di erence Sequences of Order m
4.6. The Domain of the Matrix Ar and Concerning Duality Relation
4.7. Riesz Sequence Spaces and Concerning Duality Relation
4.8. Euler Sequence Spaces and Concerning Duality Relation
4.9. Domain of the Generalized Weighted Mean and Concerning ...
4.10. Domains of Triangles in the Spaces of Strongly C1 summable ...
4.11. Characterizations of Some Other Classes of Matrix Transformations
4.12. Conclusion
Chapter 5. Spectrum of Some Particular Limitation Matrices
5.1. Preliminaries, Background and Notations
5.2. Subdivisions of the Spectrum
5.3. The Fine Spectrum of the Cesàro Operator in the Spaces c0 and c
5.4. The Fine Spectra of the Di erence Operator (1) On the Space `p
5.5. The Fine Spectra of the Di erence Operator (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
5.8. The Fine Spectra of the Operator B(r; s; t) On the Spaces `p and bvp
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
7.3. 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
Chapter 8. Sequences of Fuzzy Numbers
8.1. Introduction
8.2. Convergence of a Sequence of Fuzzy Numbers
8.3. Statistical Convergence of a Sequence of Fuzzy Numbers
8.4. The Classical Sets of Sequences of Fuzzy Numbers
8.5. Quasilinearity of the Classical Sets of Sequences of Fuzzy Numbers
8.6. Certain Sets of Sequences of Fuzzy Numbers De ned By a Modulus
8.7. Conclusion
Bibliography
List of Abbreviations and Symbols
Index
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Author: Feyzi Basar
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