Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian
© G. Ren and H. R. Malonek. 2010
Received: 28 December 2009
Accepted: 26 March 2010
Published: 26 May 2010
Let be a -invariant convex domain in including , where is a complex Coxeter group associated with reduced root system . We consider holomorphic functions defined in which are Dunkl polyharmonic, that is, for some integer . Here is the complex Dunkl Laplacian, and is the complex Dunkl operator attached to the Coxeter group , where is a multiplicity function on and is the reflection with respect to the root . We prove that any complex Dunkl polyharmonic function has a decomposition of the form , for all , where are complex Dunkl harmonic functions, that is, .
In , Aronszajn et al. indicated some applications of the Almansi formula in several complex variables. Its most eminent application is in spherical harmonic function theory [4, 5]. The polyharmonic functions have also applications in the theory of elasticity , in radar imaging , and in multivariate approximation [8, 9].
The purpose of this article is to extend Almansi's theorem to the theory of complex Dunkl harmonics. The theory of Dunkl harmonics developed by Dunkl [10–13] is an extension of the theory of ordinary harmonics. In 1989, Dunkl  constructed for each Coxeter group a family of commutative differential-difference operators , called Dunkl operators, which can be considered as perturbations of the usual partial derivatives by reflection parts. These operators step from the analysis of quantum many body system of Calogero-Moser-Sutherland type  in mathematical physics. They also have roots in the theory of special functions of several variables. With Dunkl operators in place of the usual partial derivatives, one can define the Laplacian in the Dunkl setting, which is a parametrized operator and invariant under reflection groups. These parametrized Laplacian suggests the theory of Dunkl harmonics. In , we obtained the Almansi decomposition for the real Dunkl operator. Now we continue to consider the Almansi decomposition for the complex Dunkl operator.
As a direct consequence, we will show that the Almansi Theorem implies the Gauss decomposition of the homogeneous polynomials into complex Dunkl harmonics.
Throughout this paper we let be a -invariant convex domain in including , that is, , , and for all and . This class of domain turns out to be natural for the Almansi decomposition. It is known that is a regular operator in such a domain. Namely, if , then .
Our main result is the following theorem.
3. Proof of the Main Theorem
Before proving Theorem 1.2, we need some lemmas.
Equation (3.23) follows directly from the assumption of induction.
Now we come to the proof of our main theorem.
Proof of Theorem 1.2.
We split the proof into two parts.
Thus the uniqueness follows by induction.
4. Gauss Decomposition
The remaining proof follows by induction.
The research is partially supported by the Unidade de Investigação "Matemática e Aplicações" of University of Aveiro and by the NNSF of China (no. 10771201).
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