Classical and Modern Methods in Summability (ISBN : 9780198501657)

Summability is a mathematical topic with a long tradition and many applications in, for example, function theory, number theory, and stochastics. It was originally based on classical analytical methods, but was strongly influenced by modern functional analytical methods during the last seven decades. The present book aims to introduce the reader to the wide field of summability and its applications, and provides an overview of the most important classical and modern methods used. Part I contains a short general introduction to summability, the basic classical theory concerning mainly inclusion theorems and theorems of the Silverman-Toeplitz type, a presentation of the most important classes of summability methods, Tauberian theorems, and applications of matrix methods. The proofs in Part I are exclusively done by applying classical analytical methods. Part II is concerned with modern functional analytical methods in summability, and contains the essential functional analytical basis required in later parts of the book, topologization of sequence spaces as K- and KF-spaces, domains of matrix methods as FK-spaces and their topological structure. In this part the proofs are of functional analytical nature only. Part III of the present book deals with topics in summability and topological sequence spaces which require the combination of classical and modern methods. It covers investigations of the constistency of matrix methods and of the bounded domain of matrix methods via Saks space theory, and the presentation of some aspects in topological sequence spaces. Lecturers, graduate students, and researchers working in summability and related topics will find this book a useful introduction and reference work.

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This graduate-level textbook provides a broad introduction to summability theory and develops some of its most important methodology. Distinguishing between classical ('hard') methods and modern ('soft') methods which are essentially based on analytical and functional analytic methods, respectively, chapters outline both types of methods and combinations of the two, with special attention to topics like convergence and divergence, matrixes, Tauberian theorems, applications, topological sequence spaces, consistency, Saks spaces, and bounded domains. Boos teaches mathematics at the FernUniversit<:a>t-Gesamthochschule. Cass teaches mathematics at The University of Western Ontario. Annotation c. Book News, Inc., Portland, OR (booknews.com)