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{{unreferenced|date=August 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[mathematical analysis]], many generalizations of [[Fourier series]] have proved to be useful.
They are all special cases of decompositions over an [[orthonormal basis]] of an [[inner product space]].
Here we consider that of [[square-integrable]] functions defined on an [[Interval (mathematics)|interval]] of the [[real line]], which is important, among others, for [[interpolation]] theory.


==Definition==
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]]
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Consider a set of [[square-integrable]] functions with values in <math> \mathbb{F}=\mathbb{C}\mbox{ or }\mathbb{R}</math>,
Registered users will be able to choose between the following three rendering modes:


:<math>\Phi = \{\varphi_n:[a,b]\rightarrow \mathbb{F}\}_{n=0}^\infty,</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


which are pairwise [[orthogonal]] for the [[inner product]]
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>\langle f, g\rangle_w = \int_a^b f(x)\,\overline{g}(x)\,w(x)\,dx</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


where ''w''(''x'') is a [[weight function]], and <math>\overline\cdot</math> represents [[complex conjugation]], i.e. <math>\overline{g}(x)=g(x)</math> for <math> \mathbb{F}=\mathbb{R}</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].


The '''generalized Fourier series''' of a [[square-integrable]] function ''f'': [''a'', ''b''] → <math> \mathbb{F}</math>,
==Demos==
with respect to Φ, is then


:<math>f(x) \sim \sum_{n=0}^\infty c_n\varphi_n(x),</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


where the coefficients are given by


:<math>c_n = {\langle f, \varphi_n \rangle_w\over \|\varphi_n\|_w^2}.</math>
* 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.


If Φ is a complete set, i.e., an [[orthonormal basis]] of the space of all square-integrable functions on [''a'', ''b''], as opposed to a smaller orthonormal set,
==Test pages ==
the relation <math>\sim \,</math> becomes equality in the ''[[L2 space|L²]]'' sense, more precisely modulo |·|<sub>''w''</sub> (not necessarily pointwise, nor [[almost everywhere]]).


== Example (Fourier–Legendre series) ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
The [[Legendre polynomials]] are solutions to the [[Sturm–Liouville theory|Sturm–Liouville problem]]  
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


: <math> \left((1-x^2)P_n'(x)\right)'+n(n+1)P_n(x)=0</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
and because of Sturm-Liouville theory, these polynomials are eigenfunctions of the problem and are solutions orthogonal with respect to the inner product above with unit weight. So we can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and
==Bug reporting==
 
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 .
:<math>f(x) \sim \sum_{n=0}^\infty c_n P_n(x),</math>
 
:<math>c_n = {\langle f, P_n \rangle_w\over \|P_n\|_w^2}</math>
 
As an example, let us calculate the Fourier–Legendre series for ''&fnof;''(''x'')&nbsp;=&nbsp;cos&nbsp;''x'' over [&minus;1,&nbsp;1]. Now,
 
:<math>
\begin{align}
c_0 & = \sin{1} = {\int_{-1}^1 \cos{x}\,dx \over \int_{-1}^1 (1)^2 \,dx} \\
c_1 & = 0 = {\int_{-1}^1 x \cos{x}\,dx \over \int_{-1}^1 x^2 \, dx} = {0 \over 2/3 } \\
c_2 & = {5 \over 2} (6 \cos{1} - 4\sin{1}) = {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \, dx \over \int_{-1}^1 {9x^4-6x^2+1 \over 4} \, dx} = {6 \cos{1} - 4\sin{1} \over 2/5 }
\end{align}
</math>
 
and a series involving these terms
 
:<math>c_2P_2(x)+c_1P_1(x)+c_0P_0(x)= {5 \over 2} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin{1}(1)</math>
:<math>= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}</math>
 
which differs from cos ''x'' by approximately 0.003, about&nbsp;0. It may be advantageous to use such Fourier–Legendre series since the eigenfunctions are all polynomials and hence the integrals and thus the coefficients are easier to calculate.
 
== Coefficient theorems ==
Some theorems on the coefficients ''c''<sub>''n''</sub> include:
 
===Bessel's inequality===
 
:<math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2\,dx.</math>
 
===Parseval's theorem===
 
If Φ is a complete set,
 
:<math>\sum_{n=0}^\infty |c_n|^2 = \int_a^b|f(x)|^2\, dx.</math>
 
==See also==
*[[Orthogonality]]
*[[Orthogonal function]]
*[[Eigenfunctions]]
*[[Vector space]]
*[[Function space]]
*[[Topological vector space]]
*[[Hilbert space]]
*[[Banach space]]
 
[[Category:Fourier analysis]]

Latest revision as of 22:52, 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

E=mc2


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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