Lectures on Summability
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107 A Peyerimhoff University of Marburg, Marburg/Lahn
Lectures on Summability
Springer-Verlag Berlin· Heidelberg· NewYork 1969
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
107 A Peyerimhoff University of Marburg, Marburg/Lahn
Lectures on Summability
Springer-Verlag Berlin· Heidelberg· NewYork 1969
Preface
These Lecture Notes contain the material which I covered in courses on summability held at the University of Utah. Salt Lake City. in 1967 and at the University of Marburg in 1967/68. The motivation for selecting this material was two-fold: To acquaint the students with the most important parts of the theory and to lead them to a point from which they could start a master's or Ph. D. thesis in some of the fields of my own research. Functional analysis was not a prerequisite for these courses. and so I had to exclude those subjects that need detailed knowledge of this kind. I have furnished some information on the literature in the context of the courses. but I have not aimed at a complete bibliography. In fact. this is almost impossible in view of the large number of publications in summability theory during the last seventy years. I want to express my warmest thanks to all who have helped to prepare this booklet. Part of the manuscript was read in detail by Dr. H. -H. K6r1e. who suggested numerous improvements. My thanks are also due to professors H. -E. Richert and D. H. Tucker as well as to cando math. J. BrUning. Dr. H. Fiedler. Dr. W. Miesner. and Dr. R. Trautner. A first version of these notes (covering the Utah course) was very carefully typed by Mrs. Ruth Andersen. and Mrs. Helga Runckel is responsible for the excellent typing of the manuscript in its present form. Finally lowe acknowledgement to a " silent partner". Professor W. B. Jurkat. though not actually engaged in writing the manuscript, had a great influence on it through many discussions during our collaboration. Alexander Peyerimhoff
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2_
Introductory Remarks In 1890 E. CesAro a
If
=A
n
[14]
proved the following theorem: b
and
n
=B
The assumptions of CesAro's Theorem do not imply the convergence of example
a
n ,. b n ,. (_l)n I
of the partial sums of
'" n+l c
c
(countern ). they only imply the convergence of the arithmetical means
n
We sketch a proof of CesAro's Theorem. Lemma O. 1.
If
sn -+ s • then
(We omit the simple proof.) of CesAro's Theorem.
A short calculation shows that
I + II
and it follows from Lemma O. 1. that
We have
again by Lemma O. 1. .
I
-+
BA .
- 3-
A consequence of Cesaro's Theorem and Lemma O. 1. is: If c
converges,
then
c
= AB
an = A,
(there is either convergence of
b c
n
= B,
and if
to the
n This fact was discovered earlier (in 1826) by N. H. Abel
n n "right value" , or no convergence at all).
[1] . He proved it in the following way: For
Ix
1< 1
we have (well known properties of power
series are used)
c(x)
a(x) b(x)
=
and it follows from Abel's limit theorem, that for c(x)
J
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