Banach Limit 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 1932, were scattered till now.
Sublinear functionals generating and dominating Banach Limit, unique Banach Limit (almost convergence), invariant means and invariant limits, absolute and strong almost convergence, applications to ergodicity, law of large numbers, Fourier series, uniform distribution of sequences, uniform density, core theorems, and functional Banach limits are discussed in this book. The discovery of functional analysis, such as the Hahn-Banach Theorem and the Banach-Steinhaus Theorem, helped the researchers to develop a modern, rich, and unified theory of sequence spaces by encompassing classical summability theory via matrix transformations and the topics related to sequence spaces, which arose from the concept of Banach limits, all of which are presented in this book.
The unique features of this book are as follows:
- All the results in this area which were scattered till now are in one place.
- The book is the first of its kind 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 and Ph.D. and advanced master’s students. The book is suitable for one- or two-semester course work for Ph.D. students, M.S. students in North America and Europe, and M.Phil. and master’s students in India.
Table of Contents
Introduction. Hahn-Banach Theorem and Generalizations. Banach and Generalized Limits. Almost Convergence Core Theorems. Absolute Almost Convergence. The space. Strong Almost Convergence. Functional Banach Limits. Matrix Transformations. Absolutely Almost and Strongly Almost Summable Sequences. Bibliography. Index.
Gokulananda Das is a Ph.D. and D.Sc in Mathematics , a retired Professor of Mathematics and former Vice Chancellor of Utkal University, Bhubaneswar, Odisha, India. He was the founder Professor of Mathematics at Sambalpur University , a commonwealth postdoctoral fellow at University college London, visiting Professor at University of California Santa Barbara ,USA. His research interests are in Functional Analysis, Sequence spaces and series, ergodicity,probability,fixed points and approximation. He has published around 200 research papers, guided 44 Ph. D. students, authored 18 books and 150 articles on Mathematics, Zero in Odia, Zero (the spiritual number) in English and many poems in Odia. He is in the Editorial Board in many reputed journals of Mathematics
and the founder Chief Editor of the Journal of Orissa Mathematical Society. He visited several Countries, received many awards including Samanta Chandra Sekhar award. Prof. Das took active interest and leadership in modernization of mathematics curriculum ,in the establishment of Institute of Mathematics and Applications, Bhubaneswar and in the activities of Orissa Mathematical Society.
Sudarsan Nanda is a Ph.D. and D.Sc. in Mathematics and at present the Professor of Eminence and Research Chair at Kalinga Institute of Industrial Technology (KIIT), Deemed to be University ( An Institute of Eminence), Bhubaneswar, Odisha,India. He was an Assistant Professor at Regional Engineering College ( National Institute of Technology) Rourkela and at Indian Institute of Technology (IIT) , Kharagpur. Professor at Utkal University, Bhubaneswar ,IIT Kharagpur and Vice Chancellor of North Orissa University ,Baripada, Odisha ,India. His research interests include functional analysis, Sequence Spaces optimization and fuzzy logic.Collaborated with reputed mathematicians such as B Kuttner,G Das,R N Mohapatra to name a few. He has published around 200 research papers, guided 21 Ph.D. students and authored/co-authored 45 conference papers and18 books and around 20 articles on popular Mathematics and Higher Education. He is in the Editorial Board of many reputed journals of Mathematics including Journal of Fuzzy Mathematics and Advances in Nonlinear Variational Inequalities (ANVI). He is a Fellow of the Institute of Mathematics and Applications (FIMA) (UK), former Associate of Abdus Salam International Centre for Theoretical Physics (ICTP) ,Trieste,(Italy), He was a visiting Professor at the University of Guelph ,Canada,Universities of Pisa and Milan in Italy,Visiting scientist at the University of Central Florida,Orlando ,USA ,Chinese University of Hongkong,visited University of Keiserlautern ,Germany under exchange program, received fulbright travel grant to visit USA, visited many countries , received several awards including distinguished senior scientist award from Odisha Bigyan Academy.