From: grubb@math.niu.edu (Daniel Grubb)
Subject: Re: Abel and Cesaro summability
Date: 19 Dec 2001 20:15:09 GMT
Newsgroups: sci.math.research
Summary: Abel summability implies Cesaro summability (sometimes)

>I could swear that I have read someplace, or someone has told me, that for
>bounded sequences Abel summability and Cesaro summability are equivalent. 
>Is this true?  I have hunted in G. H. Hardy and K. Knopp and elsewhere but
>can't find this explicitly stated.  I might be able to prove it, but that
>would pointless if I could find a reference.


The closest result I could find says that if \sum c_n is Abel
summable to s and the *partial sums* of \sum c_n are bounded, then
it is C_1 summable to s. This is exercise 3. in the last section of chapter
7 of Stromberg's book "An Introduction to Classical Real Analysis".
Since it is a bit unclear whether you want boundedness of the c_n
or of the partial sums, this could be what you are looking for.


--Dan Grubb