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In [[mathematics]], a '''delta operator''' is a shift-equivariant [[linear transformation|linear]] operator ''<math>\scriptstyle{ Q:\mathbb K[x] \longrightarrow \mathbb K[x] }</math>'' on the [[vector space]] of [[polynomial]]s in a variable <math> \scriptstyle x </math> over a [[field (mathematics)|field]] <math>\scriptstyle{ \mathbb K}</math> that reduces degrees by one.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


To say that <math>\scriptstyle Q</math> is '''shift-equivariant''' means that if <math>\scriptstyle{ g(x) = f(x + a)}</math>, then
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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:<math>{ (Qg)(x) = (Qf)(x + a)}.\,</math>
Registered users will be able to choose between the following three rendering modes:  


In other words, if ''<math>f</math>'' is a "'''shift'''" of ''<math>g</math>'', then ''<math>Qf</math>'' is also a shift of ''<math>Qg</math>'', and has the same "'''shifting vector'''" ''<math>a</math>''.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


To say that ''an operator reduces degree by one'' means that if ''<math>f</math>'' is a polynomial of degree ''<math>n</math>'', then ''<math>Qf</math>'' is either a polynomial of degree <math>n-1</math>, or, in case <math>n = 0</math>, ''<math>Qf</math>'' is 0.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Sometimes a ''delta operator'' is defined to be a shift-equivariant linear transformation on polynomials in ''<math>x</math>'' that maps ''<math>x</math>'' to a nonzero constant.  Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition, since shift-equivariance is a fairly strong condition.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Examples==
<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 forward [[difference operator]]
==Demos==


:: <math> (\Delta f)(x) = f(x + 1) - f(x)\, </math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:is a delta operator.


* [[Derivative|Differentiation]] with respect to ''x'', written as ''D'', is also a delta operator.
* 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.


* Any operator of the form
==Test pages ==
::<math>\sum_{k=1}^\infty c_k D^k</math>
: (where ''D''<sup>''n''</sup>(&fnof;) = &fnof;<sup>(''n'')</sup> is the ''n''<sup>th</sup> derivative) with <math>c_1\neq0</math> is a delta operator.  It can be shown that all delta operators can be written in this form.  For example, the difference operator given above can be expanded as
::<math>\Delta=e^D-1=\sum_{k=1}^\infty \frac{D^k}{k!}.</math>


* The generalized derivative of [[time scale calculus]] which unifies the forward difference operator with the derivative of standard [[calculus]] is a delta operator.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


* In [[computer science]] and [[cybernetics]], the term "discrete-time delta operator" (&delta;) is generally taken to mean a difference operator
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:: <math>{(\delta f)(x) = {{ f(x+\Delta t) - f(x) }  \over {\Delta t} }}, </math>
==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 .
: the [[Euler approximation]] of the usual derivative with a discrete sample time <math>\Delta t</math>. The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.
 
==Basic polynomials==
 
Every delta operator ''<math>Q</math>'' has a unique sequence of "basic polynomials", a [[polynomial sequence]] defined by three conditions:
 
* <math>\scriptstyle p_0(x)=1 ;</math>
* <math>\scriptstyle p_{n}(0)=0;</math>
* <math>\scriptstyle (Qp_n)(x)=np_{n-1}(x), \; \forall n \in \mathbb N.</math>
 
Such a sequence of basic polynomials is always of [[binomial type]], and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a [[Sheffer sequence]] -- a more general concept.
 
== See also ==
 
* [[Pincherle derivative]]
* [[Shift operator]]
* [[Umbral calculus]]
 
== References ==
* {{Citation | last1=Nikol'Skii | first1=Nikolai Kapitonovich | title=Treatise on the shift operator: spectral function theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-15021-5 | year=1986}}
 
 
[[Category:Linear algebra]]
[[Category:Polynomials]]
[[Category:Finite differences]]
 
[[pl:Operator delta]]

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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