Borel's Methods of Summability: Theory and Application

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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