- Research Article
- Open Access

# Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian

- Guangbin Ren
^{1}and - HelmuthR Malonek
^{2}Email author

**2010**:947518

https://doi.org/10.1155/2010/947518

© G. Ren and H. R. Malonek. 2010

**Received:**28 December 2009**Accepted:**26 March 2010**Published:**26 May 2010

## Abstract

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, .

## Keywords

- Homogeneous Polynomial
- Coxeter Group
- Reflection Group
- Taylor Formula
- Multiplicity Function

## 1. Introduction

and since the constants are solutions of , they are replaced by the solutions of the Laplace equation .

In [1], 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 [6], in radar imaging [7], 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 [10] 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 [14] 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 [15], 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.

where denotes the reflection in the hyperplane orthogonal to .

for any and .

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 .

Definition 1.1.

A holomorphic function
is *Dunkl polyharmonic* of degree
if
. If
, it is called *Dunkl harmonic* function.

Our main result is the following theorem.

Theorem 1.2.

Conversely, the sum in (1.14), with Dunkl harmonic in , defines a Dunkl polyharmonic function in of degree .

Remark 1.3.

By the Scheme in (1.4), we know that the formulae of above play the role of Taylor coefficient formulae. These formulae are new even in the classical case .

## 2. Preliminaries

Let us recall some notation in the theory of Dunkl harmonics; see [16, 17]. Concerning root system and reflection groups, see [18].

A root system is a finite set of nonzero vectors in such that and for all .

The positive subsystem is a subset of such that , where and are separated by a hyperplane through the origin.

where the symbol and .

The Coxeter group (or the finite reflection group) generated by the root system is the subgroup of the unitary group generated by .

A multiplicity function is a -invariant complex valued function defined on , that is, for all .

Notice that Dunkl operators were studied in literature for .

When , the Dunkl Laplacian is just the ordinary complex Laplacian .

Example 2.1.

where acts on by interchanging and ; more precisely, with , , and for any .

Example 2.2.

If is an even function, then the third term in the formula of vanishes, while the sum of the first two items provides a singular Sturm-Liouville operator.

## 3. Proof of the Main Theorem

Before proving Theorem 1.2, we need some lemmas.

We write instead of when .

Lemma 3.1.

Proof.

From the above two identities and the definitions of and , we have and

Lemma 3.2.

Proof.

Since , it follows that for a.e. . From the regularity property of Dunkl operators, maps into . By the continuity, Lemma 3.2 follows.

Lemma 3.3.

Proof.

Indeed, from Lemma 3.1, . As direct consequence of (3.6) and (3.11), we find that and are Dunkl harmonic, whenever is Dunkl harmonic. From the definition of , we thus obtain .

Lemma 3.4.

Proof.

Since , summing up the above identity leads to identity (3.12) for .

Lemma 3.5.

Proof.

for any Dunkl harmonic function in and .

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.

Next, since and , we have , as desired.This proves that . By induction, we can easily deduce that .

Thus the uniqueness follows by induction.

for any .

## 4. Gauss Decomposition

Corollary 4.1.

Proof.

The remaining proof follows by induction.

## Declarations

### Acknowledgment

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).

## Authors’ Affiliations

## References

- Aronszajn N, Creese TM, Lipkin LJ:
*Polyharmonic Functions*. The Clarendon Press, Oxford University Press, New York, NY, USA; 1983:x+265.MATHGoogle Scholar - Hayman WK, Korenblum B: Representation and uniqueness theorems for polyharmonic functions.
*Journal d'Analyse Mathématique*1993, 60: 113–133.MATHMathSciNetView ArticleGoogle Scholar - Malonek HR, Ren G: Almansi-type theorems in Clifford analysis.
*Mathematical Methods in the Applied Sciences*2002, 25(16–18):1541–1552.MATHMathSciNetView ArticleGoogle Scholar - Stein EM, Weiss G:
*Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no. 32*. Princeton University Press, Princeton, NJ, USA; 1971:x+297.Google Scholar - Vilenkin NJa, Klimyk AU:
*Representation of Lie Groups and Special Functions. Recent Advances, Mathematics and Its Applications*.*Volume 316*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1995:xvi+497.View ArticleGoogle Scholar - Lurie SA, Vasiliev VV:
*The Biharmonic Problem in the Theory of Elasticity*. Gordon and Breach, Luxembourg, UK; 1995:viii+265.Google Scholar - Andersson L-E, Elfving T, Golub GH: Solution of biharmonic equations with application to radar imaging.
*Journal of Computational and Applied Mathematics*1998, 94(2):153–180. 10.1016/S0377-0427(98)00079-XMATHMathSciNetView ArticleGoogle Scholar - Armitage DH, Gardiner SJ, Haussmann W, Rogge L: Characterization of best harmonic and superharmonic -approximants.
*Journal für die Reine und Angewandte Mathematik*1996, 478: 1–15.MATHMathSciNetGoogle Scholar - Kounchev O:
*Multivariate Polysplines: Applications to Numerical and Wavelet Analysis*. Academic Press, San Diego, Calif, USA; 2001:xiv+498.Google Scholar - Dunkl CF: Differential-difference operators associated to reflection groups.
*Transactions of the American Mathematical Society*1989, 311(1):167–183. 10.1090/S0002-9947-1989-0951883-8MATHMathSciNetView ArticleGoogle Scholar - Dunkl CF: Reflection groups and orthogonal polynomials on the sphere.
*Mathematische Zeitschrift*1988, 197(1):33–60. 10.1007/BF01161629MATHMathSciNetView ArticleGoogle Scholar - Dunkl CF: Integral kernels with reflection group invariance.
*Canadian Journal of Mathematics*1991, 43(6):1213–1227. 10.4153/CJM-1991-069-8MATHMathSciNetView ArticleGoogle Scholar - Dunkl CF: Intertwining operators associated to the group .
*Transactions of the American Mathematical Society*1995, 347(9):3347–3374. 10.2307/2155014MATHMathSciNetGoogle Scholar - van Diejen JF, Vinet L:
*Calogero-Sutherland-Morser Models*. Springer, New York, NY, USA; 2000.View ArticleGoogle Scholar - Ren G: Almansi decomposition for Dunkl operators.
*Science in China Series A*2005, 48(supplement 1):333–342.MATHView ArticleGoogle Scholar - Ren GB, Malonek HR: Complex Dunkl operators. http://arxiv.org/abs/0912.5196
- Dunkl CF, Xu Y:
*Orthogonal Polynomials of Several Variables*.*Volume 81*. Cambridge University Press, Cambridge, UK; 2001:xvi+390.MATHView ArticleGoogle Scholar - Humphreys JE:
*Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics*.*Volume 29*. Cambridge University Press, Cambridge, UK; 1990:xii+204.View ArticleGoogle Scholar

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