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In [[mathematics]], the '''Cesàro means''' (also called '''Cesàro averages''') of a [[sequence]] (''a''<sub>''n''</sub>) are the terms of the sequence (''c''<sub>''n''</sub>), where
In which to start, all we claim to accomplish is realize a authentic little furniture in this way having your adapted prices, as well as , again I will coming you how to take linear interpolation to aid account any added rate.<br><br>


:<math>c_n = \frac{1}{n} \sum_{i=1}^{n} a_i</math>
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is the [[arithmetic mean]] of the first ''n'' elements of (''a''<sub>''n''</sub>).
<ref name="Hardy">
{{cite book | last = Hardy | first = G. H.| title = Divergent Series | publisher = American Mathematical Society | location = Providence | year = 1992 | isbn = 978-0-8218-2649-2 }}
</ref>{{rp|96}} This concept is named after [[Ernesto Cesàro]] (1859 - 1906).
 
A basic result
<ref name="Hardy"/>{{rp|100-102}}
states that if  
:<math>\lim_{n \to \infty} a_n = A</math>
 
then also
:<math>\lim_{n \to \infty} c_n = A.</math>
 
That is, the operation of taking Cesàro means preserves [[Limit of a sequence|convergent sequence]]s and their limits. This is the basis for taking Cesàro means as a [[summability method]] in the theory of [[divergent series]].  
If the sequence of the Cesàro means is convergent, the series is said to be ''Cesàro summable''.  There are certainly many examples for which the sequence of Cesàro means converges, but the original sequence does not: for example with
 
: <math>a_n=\begin{cases}1&\mbox{if }n=2k-1, \\ 0&\mbox{if }n=2k\end{cases}</math>,
 
we have an [[oscillating sequence]], but the means have limit <math>\frac{1}{2}</math>. (See also [[Grandi's series]].)
 
Another example is the sequence <math>a_n=(-1)^n</math> which is Cesàro summable to <math>1/2</math> and has Cesàro-mean <math> 0 </math>.
 
Cesàro means are often applied to [[Fourier series]],
<ref name="Katznelson">
{{cite book | last = Katznelson | first = Yitzhak | title = An Introduction to Harmonic Analysis | publisher = Dover Publications | location = New York | year = 1976 | isbn = 978-0-486-63331-2 }}
</ref>{{rp|11-13}}
since the means (applied to the [[trigonometric polynomial]]s making up the symmetric [[partial sum]]s) are more powerful in summing such series than [[pointwise convergence]]. The kernel that corresponds is the [[Fejér kernel]], replacing the [[Dirichlet kernel]]; it is positive, while the Dirichlet kernel takes both positive and negative values. This accounts for the superior properties of Cesàro means for summing Fourier series, according to the general theory of [[approximate identity|approximate identities]].
 
A generalization of the Cesàro mean is the [[Stolz-Cesàro theorem]].
 
The [[Riesz mean]] was introduced by [[M. Riesz]] as a more powerful but substantially similar [[summability method]].
 
== See also ==
* [[Cesàro summation]]
 
== References ==
{{Reflist}}
 
==External links==
*[http://planetmath.org/encyclopedia/CesaroMean2.html Cesaro mean at PlanetMath]
*[http://www.sosmath.com/calculus/sequence/hardlim/hardlim.html Cesaro mean at SOS Math]
 
{{DEFAULTSORT:Cesaro mean}}
[[Category:Means]]
[[Category:Mathematical series]]
 
[[de:Cesàro-Mittel]]

Latest revision as of 00:04, 29 December 2014

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