Classical and Modern Methods in Summability

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Oxford University Press, 2000 - Mathematics - 586 pages
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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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Contents

CLASSICAL METHODS IN SUMMABILITY
1
basic classical theory
26
Special summability methods
99
Tauberian theorems
167
Application of matrix methods
205
FUNCTIONAL ANALYTIC METHODS
259
K and FKspaces
338
structure of the domains
396
FUNCTIONAL ANALYTIC METHODS
457
Saks spaces and bounded domains
515
Some aspects of topological sequence spaces
538
Bibliography
563
Index
575
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About the author (2000)

J. Boos, Professor of Mathematics, FernUniversitšt-Gesamthochschule Hagen, Germany.

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