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| In [[mathematics]], particularly the study of [[Lie groups]], a '''Dunkl operator''' is a certain kind of [[mathematical operator]], involving [[differential operator]]s but also [[Reflection (mathematics)|reflection]]s in an underlying space.
| | Friends call him Royal. What she loves performing is playing croquet and she is trying to make it a profession. The job I've been occupying for many years is a bookkeeper but I've already applied for another 1. Her family members life in Delaware but she requirements to move simply because of her family.<br><br>My weblog: [http://Www.sethtemple.org/UserProfile/tabid/42/userId/161658/Default.aspx Www.sethtemple.org] |
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| Formally, let ''G'' be a [[Coxeter group]] with reduced root system ''R'' and ''k''<sub>''v''</sub> a multiplicity function on ''R'' (so ''k''<sub>''u''</sub> = ''k''<sub>''v''</sub> whenever the reflections σ<sub>''u''</sub> and σ<sub>''v''</sub> corresponding to the roots ''u'' and ''v'' are conjugate in ''G''). Then, the '''Dunkl operator''' is defined by:
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| :<math>T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i</math>
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| where <math>v_i </math> is the ''i''-th component of ''v'', 1 ≤ ''i'' ≤ ''N'', ''x'' in ''R''<sup>''N''</sup>, and ''f'' a smooth function on ''R''<sup>''N''</sup>.
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| Dunkl operators were introduced by {{harvs|txt|authorlink=Charles F. Dunkl|last=Dunkl|year=1989}}. One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy <math>T_i (T_j f(x)) = T_j (T_i f(x))</math> just as partial derivatives do. Thus Dunkl operators represent a meaningful generalization of partial derivatives.
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| ==References==
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| *{{Citation | last1=Dunkl | first1=Charles F. | title=Differential-difference operators associated to reflection groups | doi=10.2307/2001022 | mr=951883 | year=1989 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=311 | issue=1 | pages=167–183}}
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| [[Category:Lie groups]]
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Latest revision as of 00:59, 6 November 2014
Friends call him Royal. What she loves performing is playing croquet and she is trying to make it a profession. The job I've been occupying for many years is a bookkeeper but I've already applied for another 1. Her family members life in Delaware but she requirements to move simply because of her family.
My weblog: Www.sethtemple.org