Summability methods are transformations that map sequences (or functions) to sequences (or functions). A prime requirement for a "good" summability method is that it preserves convergence. Unless it is the identity transformation, it will do more: it will transform some divergent sequences to convergent sequences.
An important type of theorem is called a Tauberian theorem. Here, we know that a sequence is summable. The sequence satisfies a further property that implies convergence.
Borel's methods are fundamental to a whole class of sequences to function methods. The transformation gives a function that is usually analytic in a large part of the complex plane, leading to a method for analytic continuation.
These methods, dated from the beginning of the 20th century, have recently found applications in some problems in theoretical physics.
Readership: Research mathematicians.
Author(s): Bruce Shawyer, Bruce Watson
Series: Oxford Mathematical Monographs
Publisher: Oxford University Press
Year: 1994
Language: English
Pages: C, xii+242
Cover
Series Editors
Publications List of OXFORD MATHEMATICAL MONOGRAPHS
Borel's Methods of Summability: Theory and Applications
© Bruce Shawyer and Bruce Watson, 1994
ISBN 0198535856 /9780198535850
Dedication
Preface
Acknowledgement
Contents
0 Introduction
1 HISTORICAL OVERVIEW
2 SUMMABILITY METHODS IN GENERAL
3 BOREL'S METHODS OF SUMMABILITY
4 RELATIONS WITH THE FAMILY OF CIRCLE METHODS
5 GENERALIZATIONS OF BOREL'S METHODS
6 ABELIAN THEOREMS.
7 and 8 TAUBERIAN THEOREMS I AND II
9 RELATIONSHIPS WITH OTHER METHODS
10 APPLICATIONS OF BOREL'S METHODS
11 REFERENCES
1 Historical overview
2 Summability methods in general
2.1 Regularity
2.2 Generalized Cesaro summability
2.3 Sequence-to-function methods based on power series
3 Borel's methods of summability
3.1 Basic definitions
3.2 Basic properties of Borel's methods
3.2.1 Common properties
3.2.2 Interrelationships
3.3 Extensions
3.3.1 Absolute summability
3.3.2 Strong summability
3.3.3 Normal and regular summability
3.4 Relationships with other methods
3.4.1 Euler methods
3.4.2 Cesaro and Abel methods
3.4.3 Other methods
3.5 Abelian theorems
4 Relations with the family of circle methods
4.1 Euler-Knopp summability methods
4.2 Ta methods
4.2.1 Definitions
4.2.2 Ta on ser
4.2.3 Taand Tb
4.2.4 Ta and B
4.2.5 Translativity
4.3 Meyer-Konig's Sa methods
4.3.1 Definition
4.3.2 Translativity
4.3.3 Sa on serie
4.3.4 Sa and So
4.3.5 Sa and Ep
4.3.6 Function theoretic considerations
4.4 Relations of Ta and Sa with Ep and B
4.5 Relations of Ep, B, and Sa with Ta
4.6 Equivalence of Ep, B, Sa, Ta for bounded sequences
4.7 Tauberian theorems
5 Generalizations of Borel's methods
5.1 First attempts
5.2 Mittag-Lefer's functions
5.3 Borel-type methods
5.3.1 Definitions
5.3.2 Preliminaries
5.3.3 Lemmas
5.4 Relationships with respect to the parameter a
5.5 Abelian relationships with respect to the parameter ß
5.5.1 Interrelationships with same type
5.5.2 Interrelationships between types
5.6 Tauberian relationships with respect to the parameter ß
5.6.1 Preliminary results
5.6.2 Proofs of the theorems
5.7 Extended definitions
5.7.1 Results involving strong summabilit
5.7.2 Results involving absolute summability
6 Abelian theorems
6.1 Introduction
6.2 Abelian theorems for ordinary Borel-type methods
6.3 Abelian theorems for strong Borel-type methods
6.4 Abelian theorems for absolute Borel-type methods
7 Tauberian theorems - I
7.1 The `o' theorem
7.1.1 Preliminary results
7.1.2 Results on Cesaro sums
7.1.3 Proof of the `o' theorem
7.2 The `0' theorem
7.2.1 Preliminary results
7.2.2 Estimates of some sums as integrals
7.2.3 Results on summability (e, c)
7.2.4 Two preliminary theorems
7.2.5 Proof of the `0' theorem
7.3 Kwee's `0' theorem
7.3.1 Preliminary results
7.3.2 Proof of Kwee's `O' theorem
7.3.3 Kwee's `O' theorem is best possible
8 Tauberian theorems - II
8.1 The slowly decreasing theorem
8.1.1 Preliminary results
8.2 An equivalence theorem
8.3 Proof of the slowly decreasing theorem
8.4 Gap theorems
9 Relationships with other methods
9.1 Product methods with the Cesaro method
9.1.1 Product methods
9.1.2 Preliminary results
9.1.3 Proof of the Cesaro product theorem
9.2 Abelian relations with the Abel-type methods
9.2.1 Review of the definitions
9.2.2 Preliminary results
9.2.3 Theorems from Borel to Abel
9.3 Tauberian relations with the Abel-type methods
9.3.1 Preliminary results
9.3.2 Theorems from Abel to Borel
9.4 Tauberian relations with the logarithmic method
9.4.1 Preliminary results
9.4.2 The logarithmic theorem
9.5 Relations with the Lambert method
9.5.1 Transformation formulae
9.5.2 Essential lemmas
9.5.3 Proof of the Lambert theorem
10 Applications of Borel's methods
Borel's methods in mathematics
10.1 An early application
10.2 Laplace transforms
10.3 Entire functions and the Borel transform
10.3.1 The Phragmen-Lindelof indicator function
10.3.2 The conjugate indicator diagram
10.4 Arithmetical functions
Recent applications in mathematical physics
10.5 Basic theory
References
Bibliography
TEXTBOOKS, THESES, LECTURE NOTES
ARTICLES
APPLIED ARTICLES
Index
