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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
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


:<math>c_n = \frac{1}{n} \sum_{i=1}^{n} a_i</math>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


is the [[arithmetic mean]] of the first ''n'' elements of (''a''<sub>''n''</sub>). 
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<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
'''MathML'''
<ref name="Hardy"/>{{rp|100-102}}
:<math forcemathmode="mathml">E=mc^2</math>
states that if
:<math>\lim_{n \to \infty} a_n = A</math>


then also
<!--'''PNG'''  (currently default in production)
:<math>\lim_{n \to \infty} c_n = A.</math>
:<math forcemathmode="png">E=mc^2</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]].
'''source'''
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 forcemathmode="source">E=mc^2</math> -->


: <math>a_n=\begin{cases}1&\mbox{if }n=2k-1, \\ 0&\mbox{if }n=2k\end{cases}</math>,
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


we have an [[oscillating sequence]], but the means have limit <math>\frac{1}{2}</math>. (See also [[Grandi's series]].)
==Demos==


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>.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


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]].
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


The [[Riesz mean]] was introduced by [[M. Riesz]] as a more powerful but substantially similar [[summability method]].
==Test pages ==


== See also ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
* [[Cesàro summation]]
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


== References ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
{{Reflist}}
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
==External links==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*[http://planetmath.org/encyclopedia/CesaroMean.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 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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