Guangbin Ren

Clifford analysis for finite reflection groups

In this article we establish the basis for a Clifford analysis over finite reflection groups. §Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Almansi decomposition for Dunkl operators

AbstractLet Ω be a G-invariant convex domain in ℝN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (Δh)nf = 0 for some integer n. Here333-01is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G,

(DISCRETE) ALMANSI TYPE DECOMPOSITIONS: AN UMBRAL CALCULUS FRAMEWORK BASED ON osp(1|2) SYMMETRIES

\mathcal{D}_j f(x) = \frac{\partial }{{\partial x_j }}f(x) + \sum\limits_{v \in R_ + } {\kappa _v \frac{{f(x) - f(\sigma _v x)}}{{\left\langle {x,v} \right\rangle }}} v_j ,

Holomorphic Jackson's theorems in polydiscs

where Kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form

Weighted Hölder Continuity of Hyperbolic Harmonic Bloch functions

f(x) = f_0 (x) + \left| x \right|^2 f_1 (x) + \cdots + \left| x \right|^{2(n - 1)} f_{n - 1} (x), \forall x \in \Omega ,

Almansi Decomposition for Dunkl-Helmholtz Operators

where fj are Dunkl harmonic functions, i.e. Δhfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.