From: bruck@math.usc.edu (Ronald Bruck) Subject: Re: sequence problem---help needed Date: Mon, 28 Jun 1999 22:34:56 -0700 Newsgroups: sci.math Keywords: (Tauberian) types of convergence of sequences In article <290619990938111182%fly@spam.dialix.com.au>, wrote: :Any help with the following problem would be greatly appreciated: : :Suppose we have a countably infinite set of numbers x_1, x_2, ... . :Each of these numbers is a real in the interval [0,1]. Consider the :sequence: :x_1/1, (x_1 + x_2)/2, (x_1 + x_2 + x_3)/3, ... : :Is it possible for this sequence to diverge? : :Suppose we order the reals in the set differently: `x_1' in the :definition of the sequence still denotes the first member of the set :and `x_2' the second member, etc., but these are now different numbers. :Is it possible for the sequence to converge to different limits :relative to different orderings, or to converge relative to some :orderings and diverge relative to others? Yes, it can diverge. Start the sequence with a 0. Now take a long string of 1's, until the average is close to 1. Now take a long string of 0's, until the average is close to 0. Now take a long string of 1's, until the average is close to 1 again, in fact within half the distance of 1 it was the last time it was near 1. Now take a long string of 0's, until the average is close to 0 again, in fact within half the distance of 0 it was the last time it was near 0. Keep doing this. Then there is a subsequence which converges to 0, and a subsequence which converges to 1. In fact, given ANY real number between 0 and 1, there's a subsequence which converges to that real number. (This is because s_n = (x_1 + ... + x_n)/n has the property that s_{n+1} - s_n --> 0. It's true of any sequence in a compact metric space for which d(s_n,s_{n+1}) tends to 0 that the subsequential limits form a connected set.) Note that this sequence can be rearranged so the averages converge to 1 alone; just put the n-th 0 at the n! position, and make everything else a 1. In fact it can be rearranged so the lim inf of the averages is any fixed a in [0,1], and the lim sup is any fixed b in [a,1]. And as I just remarked, everything in between will also be a subsequential limit. There IS a Tauberian theorem related to this question. If the shifted averages (x_{p+1} + x_{p+2} + ... + x_{p+n})/n converge as n --> infinity, UNIFORMLY in p >= 0, and if x_n - x_{n+1} --> 0, then {x_n} converges. The Cesaro convergence, uniform in the shift, is sometimes called "almost convergence" (although, AFAIK, it has nothing to do with "almost-everywhere convergence", except that it is related to mean ergodic theorems). --Ron Bruck ============================================================================== From: Panayiotis Spanos Subject: Re: Tauberian/Abelian Theorems/Constants Date: Fri, 10 Dec 1999 02:29:44 +0000 Newsgroups: sci.math.research Strongly recommended for a simple quick intro into summability is Powell and Shah's Summability theory and applications (Van Nostrand Reinhold) (1972) and specifically on Tauberian theory J. van de Lune's An introduction to Tauberian theory: from Tauber to Wiener (CWI Syllabus 12) as well as A. Peyerimhoff's Lectures on Summability (Springer-Verlag 107) Regards Pi Simon Langley wrote: > Help! > > I've just agreed to supervise an undergraduate student whose tutor has > been taken ill. The student's project is to rework some calculations > of 'Tauberian Constants connected with the Prime Number Theorem' - not > something I know much about (nor, alas, does anyone else in this > department). Though we probably understand enough to do the > calculations, we'd both like to have a bit more idea of the background. > > Apostol's Intro to Analytic Number Theory was quite helpful but does > anyone know any introductory sources for: Tauberian Theorems, Abelian > transforms, Ingham summability .... We have a few references to papers > (mostly pre 1970) and some later ones which no undergraduate is likely > to follow. > > Any suggestions much appreciated. > > Thanks, > > Simon Langley > University of the West of England -- http://www.geocities.com/Athens/Styx/3079/