# 1014 a find the jordan form for a b for any function f

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Unformatted text preview: ar sequence {α1 , α2 , α3 , . . .} there is an associated sequence of averages {µ1 , µ2 , µ3 , . . .} in which Y P µ1 = α1 , µ2 = α1 + α2 , 2 ..., µn = α1 + α2 + · · · + αn . n 85 This sequence of averages is called the associated Ces`ro sequence, and when a a limn→∞ µn = α, we say that {αn } is Ces`ro summable (or merely summable ) to α. It can be proven (Exercise 7.10.11) that if {αn } converges to α, then {µn } converges to α, but not conversely. In other words, convergence implies summability, but summability doesn’t insure convergence. To see that a sequence can be summable without being convergent, notice that the oscillatory sequence {0, 1, 0, 1, . . .} doesn’t converge, but it is Ces`ro summable to 1/2, which is the a mean value of {0, 1}. This is typical because averaging has a smoothing eﬀect so that oscillations that prohibit convergence of the original sequence tend to be smoothed away or averaged out in the Ces`ro sequence. a O C 85 Ernesto Ces`ro (1859–1906) was an Italian mathematician who worked mainly in diﬀerential a geometry but also contributed to number theory, divergent series, and mathematical physics. After studying in Naples, Li`ge, and Paris, Ces`ro received his doctorate from the University e a of Rome in 1887, and he went on to occupy the chair of mathematics at Palermo. Ces`ro’s a most important contribution is considered to be his 1890 book Lezione di geometria intrinseca , but, in large part, his name has been perpetuated because of its attachment to the concept of Ces`ro summability. a Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 7.10 Diﬀerence Equations, Limits, and Summability http://www.amazon.com/exec/obidos/ASIN/0898714540 631 Similar statements hold for general sequences of vectors and matrices (Exercise 7.10.11), but Ces`ro summability is particularly interesting when it is a applied to the seq...
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## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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