Cesaro means for $\alpha<1$ and Banach limits - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:22:10Z http://mathoverflow.net/feeds/question/93620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93620/cesaro-means-for-alpha1-and-banach-limits Cesaro means for $\alpha<1$ and Banach limits kap44 2012-04-10T02:21:04Z 2012-04-10T10:24:00Z <p>I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all <a href="http://en.wikipedia.org/wiki/Almost_convergent_sequence" rel="nofollow">almost convergent sequences</a>. For the Cesaro summation method $(C, 1)$ <a href="http://mathoverflow.net/questions/93516/cesaro-means-and-banach-limits" rel="nofollow">this fails</a>; is this true, e.g., for the <a href="http://en.wikipedia.org/wiki/Cesaro_summation#.28C.2C_.CE.B1.29_summation" rel="nofollow">Cesaro methods</a> $(C, \alpha)$ with $\alpha&lt;1$?</p> http://mathoverflow.net/questions/93620/cesaro-means-for-alpha1-and-banach-limits/93640#93640 Answer by Martin Sleziak for Cesaro means for $\alpha<1$ and Banach limits Martin Sleziak 2012-04-10T10:03:57Z 2012-04-10T10:24:00Z <p>The paper G.G. Lorentz: A contribution to the theory of divergent sequences; Acta mathematica, Volume 80, Number 1, 1960, 167-190; DOI: <a href="http://dx.doi.org/10.1007/BF02393648" rel="nofollow">10.1007/BF02393648</a>, contains several interesting results related to your questions on Banach limits.</p> <p>A characterization of matrix methods that sum all almost convergent sequences is given (Theorem 7). In particular, each $C_\alpha$ sums all almost convergent sequences.</p> <p>However, it is shown that almost convergence cannot be represented by a regular matrix method. Also the following stronger result about a class of matrix methods is shown.</p> <p><strong>Theorem 11.</strong> For every sequence ${A_k}$ of methods of the class $\mathfrak A$ there is a bounded sequence $x = {x_n}$ which is not almost convergent but is summable to the value zero by every one of the methods $A_k$.</p> <p>The class $\mathfrak A$ in Lorentz's paper is the class of matrices fulfilling $$\lim\limits_{m\to\infty} \max\limits_n |a_{mn}|=0.$$ I think it's not that hard to show that each matrix $C_n$ belongs to $\mathfrak A$.</p> <p>Some further references for almost convergence are mentioned e.g in the book Boos: <a href="http://books.google.com/books?id=kZ9cy6XyidEC" rel="nofollow">Classical and Modern Methods in Summability</a>.</p>