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Dunkl operator

In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space.

Formally, let G be a Coxeter group with reduced root system R and k_v a multiplicity function on R (so k_u = k_v whenever the reflections σ_u and σ_v corresponding to the roots u and v are conjugate in G). Then, the Dunkl operator is defined by:

$T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i$

for 1 <= i <= N, x ∈ R^N, and f a smooth function on R^N.

Dunkl operators were first considered by mathematician Charles F. Dunkl .

Categories: Lie groups

Last updated: 10-12-2005 15:12:58

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10-26-2009 08:16:03
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Dunkl operator