Banach Limit and Applications

Description

Banach Limits and Applications provides all the results in the area of Banach limit, its extensions, generalizations and applications to various fields in one go (as far as possible). All the results in this field, after Banach introduced this concept in the year 1932 , are scattered till now. Sublinear functionals generating and dominating Banach Limit, unique Branch Limit (almost convergence), invariant means and invariant limits, absolute and strong almost convergence, applications to ergodicity, law of large number, Fourier series, uniform distribution of sequences, uniform density, core theorems, functional Banach limits are discussed in this book. Discovery of functional analysis such as Hahn-Banach theorem, Banach-Steinhaus Theorem helped the researchers to develop a modern, rich and unified theory of sequence spaces by enveloping classical summability theory via matrix transformation and the topics related to sequence spaces arose from the concept of Banach limit are presented in this book. The unique features of this book are as follows: It contains all the results in this area at one place which are scattered till now. The book is first of its kinds in the sense that there is no other competitive book . The contents of this monograph did not appear in any book form before. The audience of this book are the researchers in this area, the Ph.D. and advanced Masters students. The book is suitable for one or two semester course work for Ph. D. students, M.S. Students of North America and Europe, M. Phil and Masters Students of India. Cover Half Title Title Page Copyright Page Contents Preface Authors 1. Introduction 2. Hahn-Banach Theorem and Generalizations 2.1. Extended form of Hahn-Banach Theorem 2.1.1. Discussion 2.2. Invariant linear functionals 2.3. Hahn-Banach Theorem for semifields 2.4. Sandwich version of Hahn-Banach Theorem 2.5. Hahn-Banach Theorem for modules 3. Banach and Generalized Limits 3.1. Banach limits 3.2. Sublinear functionals generating Banach limits 3.3. Sublinear functional that dominates all Banach limits 3.4. Sublinear functionals involving regular matrices 3.5. Invariant limits 3.6. Generalizations of Banach limit 3.7. Invariant means 3.8. Further generalization 4. Almost Convergence 4.1. Unique Banach limits 4.2. Properties and examples of almost convergent sequences 4.2.1. Examples of almost convergent sequences 4.2.2. Periodic sequence 4.3. Almost convergence and matrix method 4.4. Almost convergence and uniform distribution of sequences 4.4.1. Three parameter matrices 4.5. Well-distributed sequences 4.6. Almost convergence and statistical convergence 4.6.1. Properties 4.7. Almost convergence, density, and uniform density 4.8. Almost convergence and uniform law of large numbers 4.9. Almost convergence and almost ergodicity 5. Core Theorems 5.1. Knopp's Core 5.2. Banach Core 5.3. Generalized CORE theorems 5.4. Main theorems 6. Absolute Almost Convergence 6.1. Introducing the new sequence space 6.2. Emerging eight spaces 6.3. Theorems 6.4. Application of absolute almost convergence to Fourier series 6.5. Tauberian Theorem 7. The Space ^l (p) 7.1. New sequence Spaces ^l(p) and ^l(p) 7.1.1. An important property 7.2. Some topological properties 7.3. Some further properties 7.4. Some further topological properties 7.5. Some outstanding conjectures 7.6. Absolute Banach limits 8. Strong Almost Convergence 8.1. Strong almost convergence 8.2. Strong 􀀀 almost convergence 8.3. On lacunary strong almost convergence 9. Functional Banach Limits 9.1. Preliminaries 9.2. Functional Banach limits 9.3. The sets S and jSj 9.4. Absolute almost convergence 9.5. Strong almost convergence 10. Matrix Transformations 10.1. Introduction 10.2. The classes (`1(p); ^c) and (c(p)^c) 10.3. The classes (`(p); ^c) 10.4. The spaces ^c0(p) and ^c(p) 10.5. The classes (v; ^c) 10.6. The class (^c; c) 10.7. The class (`1; ^ `p) 10.8. The class (`1; ^ `p) 10.9. The class (`p; ^ `p) 10.10. Some open problems 11. Absolutely Almost and Strongly Almost Summable Sequences 11.1. Introduction 11.2. Absolute almost summability 11.3. Strong almost summability 11.4. Strong almost (Z; k) summability 11.5. Spaces of strongly almost summable sequences Bibliography Index