Volume 2010, Article ID 947518,13pages doi:10.1155/2010/947518
Research Article
Decomposition of Polyharmonic Functions with
Respect to the Complex Dunkl Laplacian
Guangbin Ren
1and Helmuth R. Malonek
21Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China 2Departamento de Matem´atica, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Correspondence should be addressed to Helmuth R. Malonek,[email protected]
Received 28 December 2009; Accepted 26 March 2010
Academic Editor: Yuming Xing
Copyrightq2010 G. Ren and H. R. Malonek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
LetΩbe aG-invariant convex domain inCNincluding 0, whereGis a complex Coxeter group associated with reduced root systemR⊂RN. We consider holomorphic functionsfdefined inΩ which are Dunkl polyharmonic, that is,Δhnf 0 for some integern. HereΔh Nj1D2j is the complex Dunkl Laplacian, andDjis the complex Dunkl operator attached to the Coxeter groupG, Djfz ∂f/∂zjzv∈Rκvfz−fσvz/z, vvj,whereκvis a multiplicity function onR
andσvis the reflection with respect to the rootv. We prove that any complex Dunkl polyharmonic functionfhas a decomposition of the formfz f0zNn1zj2f1z· · ·Nn1z2j
n−1f n−1z, for allz∈Ω, wherefjare complex Dunkl harmonic functions, that is,Δhfj0.
1. Introduction
A fundamental result in the theory of polyharmonic functions is the celebrated Almansi theorem1–3 , which shows that for any polyharmonic functionf of degreenin a starlike domainDinRNwith center 0, that is,
ΔRnf : ⎛ ⎝N
j1 ∂2 ∂x2
j ⎞ ⎠ n
, f 0, 1.1
there exist uniquely harmonic functionsf0, . . . , fn−1such that
The Almansi formula is a genuine analogy to the Taylor formula:
ft f0 tf
0
1! · · ·t
nfn0
n! · · ·. 1.3
Compared with the Taylor formula, the Almansi formula is obtained by the scheme
d
dt −→ΔR, 1.4
and since the constantsfn0/n! are solutions ofd/dtfn0/n! 0, they are replaced
by the solutions of the Laplace equationΔRfk0.
In1 , Aronszajn et al. indicated some applications of the Almansi formula in several complex variables. Its most eminent application is in spherical harmonic function theory4,
5 . The polyharmonic functions have also applications in the theory of elasticity6 , in radar imaging7 , and in multivariate approximation8,9 .
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 Dj, 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 type14 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.
We need some notations before stating our main result. LetRbe a root system inRN andGthe associated Coxeter group. Letκ:R → Cbe a fixed multiplicity functionv→κv
onR. Fix a positive subsystemRofR, and denoteγ γκ:
v∈Rκv. We will always assume
that
Reγκ>−
N
2 . 1.5
LetDjbe the Dunkl operator attached to the Coxeter groupGand the multiplicity function
κ, defined bysee16
Djfz
∂f ∂zj
z
v∈R κv
fz−fσvz
z, v vj, 1.6
The Dunkl operators enjoy the regularity property: if f ∈ HΩ, the space of holomorphic functions inΩ, thenDif∈HΩ. This follows immediately from the formula
fz−fσvz
z, v
1
0
∇ftσvz 1−tz, 2v |v|2
dt 1.7
for anyf∈HΩandv∈R.
The Dunkl Laplacian is defined as
ΔhD12· · ·D2N, 1.8
more precisely,
Δhfz Δfz 2
v∈R κv
∇fz, v z, v −2
v∈R κv
fz−fσvz z, v2 |v|
2.
1.9
HereΔand∇are the complex Laplacian and gradient operator:
Δ: ΔC ∂
2
∂z2 1
· · · ∂2
∂z2
n
,
∇
∂ ∂z1
, . . . , ∂ ∂zn
.
1.10
Throughout this paper we letΩbe aG-invariant convex domain inCNincluding 0, that is,
GΩ⊂Ω, 0∈Ω, andtx1−ty∈Ωfor allt∈0,1 andx, y∈Ω. This class of domain turns out to be natural for the Almansi decomposition. It is known thatΔhis a regular operator in
such a domain. Namely, iff∈HΩ, thenΔhf∈HΩ.
Definition 1.1. A holomorphic function f : Ω → C is Dunkl polyharmonic of degree n if
Δhnf 0. Ifn1, it is calledDunkl harmonicfunction.
LetI be the identity operator. For anys ∈ C with Res > 0 we define the operator
Is:HΩ → HΩby
Isfx 1
0
ftxts−1dt. 1.11
If f is Dunkl harmonic in Ω, then so is Isf. For any j ∈ N, by assumption 1.5 we can
introduce the operator:
Qj
1
For anyz∈CNandj∈N, we denote
z2z21· · ·z2N, z2jz2jz21· · ·z2Nj. 1.13
Our main result is the following theorem.
Theorem 1.2. Assume thatRis a root system inRNandGits associated complex Coxter group. Let Ωbe aG-invariant convex domain inCNincluding0. Iffis a Dunkl polyharmonic function inΩof
degreen, then there exist uniquely Dunkl harmonic functionsf0, . . . , fn−1such that
fz f0z z2f1z · · ·z2n−1fn−1z, ∀x∈Ω. 1.14
Moreover the Dunkl harmonic functionsf0, . . . , fn−1are given by the following formulae:
f0I−z2Q1Δh
I−z4Q2Δ2h· · ·I−z2n−1Qn−1Δnh−1
fz
f1Q1Δh
I−z4Q2Δ2h· · ·I−z2n−1Qn−1Δnh−1
fz
.. .
fn−2Qn−2Δnh−2
I−z2n−1Qn−1Δnh−1
fz
fn−1Qn−1Δnh−1fz.
1.15
Conversely, the sum in 1.14, with f0, . . . , fn−1 Dunkl harmonic in Ω, defines a Dunkl
polyharmonic function inΩof degreen.
Remark 1.3. By the Scheme in1.4, we know that the formulae offj above play the role of
Taylor coefficient formulae. These formulae are new even in the classical caseκ0.
2. Preliminaries
Let us recall some notation in the theory of Dunkl harmonics; see16,17 . Concerning root system and reflection groups, see18 .
A root systemRis a finite set of nonzero vectors inRNsuch thatσ
vRRandR∩Rv {±v}for allv∈R.
The positive subsystemR is a subset ofR such thatR R∪−R, whereRand −Rare separated by a hyperplane through the origin.
For a nonzero vectorv ∈ CN, the reflectionσ
v in the hyperplane orthogonal tovis
defined by
σvz:z−2
z, v
|v|2 v, z∈C
N,
2.1
The Coxeter groupGor the finite reflection groupgenerated by the root systemRis the subgroup of the unitary groupUNgenerated by{σu:u∈R}.
A multiplicity functionκvis aG-invariant complex valued function defined onR, that
is,κvκgvfor allg∈G.
Notice that Dunkl operators were studied in literature for Reκv ≥0.
The Dunkl operator Dj, associated with the Coxeter group G and the multiplicity
function κ, is the first-order differential-difference operator. The remarkable property of Dunkl operators is that they are commutative:
DiDjDjDi. 2.2
The Dunkl LaplacianΔhNj1Dj2can be split into three parts
Δh Δ GhDh 2.3
with
Ghfz 2
v∈R κv∇
fz, v z, v ;
Dhfz −2
v∈R κv
fz−fσvz z, v2 |v|
2.
2.4
Whenκ0, the Dunkl LaplacianΔhis just the ordinary complex LaplacianΔ.
Consider the natural action ofUN on functionsf : CN → C, given bygfz
fg−1z. The Dunkl LaplacianΔ
hisG-invariant, that is,
g◦Δh Δh◦g, ∀g ∈G. 2.5
Example 2.1. LetN be an integer andN ≥ 2. Since we need to consider the sumi /j andi
runs from 1 toN, this forcesN ≥2.Take the Coxeter groupGSN, which is the symmetric
group inNelements, acting onRNby permuting the standard basise1, . . . , eNsee17, page
289 . We regard the transpositionijinSNas a reflectionσijsuch that
σij
ei−ej
−ei−ej
. 2.6
Therefore,SNis a finite reflection generated byσijwith a root system
R±ei−ej
: 1≤i < j≤N. 2.7
As all transpositions are conjugate in SN, the vector space of multiplicity function is one
dimensional. The complex Dunkl operators associated with the multiplicity parametersκ∈C
are given by
Djfz
∂f ∂zj
z κ
i /j
fz−fσijz
zi−zj
whereσijacts onCNby interchangingziandzj; more precisely,σijz σijz1, . . . , σijznwith
σijzizj,σijzjzi, andσijzkzkfor anyk /i, j.
In this case, the condition1.5of the main theorem reduces to
κ >− 1
N−1. 2.9
Example 2.2. In the one-dimensional caseN1, the root systemRis of typeA1, the reflection
groupGZ2, and the multiplicity function is given by a single parameterκ∈C. The Dunkl
operatorD:D1and the Dunkl LaplacianΔhare given, respectively, by
Dfz fz κfz−f−z
z ,
Δhfz fz 2κ
fz z −2κ
fz−f−z z2 .
2.10
Iffis an even function, then the third term in the formula ofΔhfvanishes, while the
sum of the first two items provides a singular Sturm-Liouville operator.
3. Proof of the Main Theorem
Before provingTheorem 1.2, we need some lemmas. Denote
RssI N
j1 zj ∂
∂zj
. 3.1
We writeRinstead ofR0whens0.
Lemma 3.1. Ifs∈C,Res >0, andf∈HΩ, then
fz IsRsfz RsIsfz. 3.2
Proof. For anys∈C, Res >0 andf∈HΩ,
fz
1
0 d dt
tsftzdt. 3.3
By direct calculation
d dt
tsftzsts−1ftz ts−1
N
i1 wi
∂f ∂wi
wherewitzi. Therefore
fz
1
0
sftz
N
i1 wi ∂f ∂wi tz
ts−1dt;
fz s
1
0
ftzts−1dt
N
i1 wi
∂ ∂wi
1
0
ftzts−1dt.
3.5
From the above two identities and the definitions ofIsandRs, we havefz IsRsfzand
fz RsIsfz.
Lemma 3.2. Iff ∈HΩ, then for anys∈C,Res >0, andz∈Ω
ΔhIsfz Is2Δhfz. 3.6
Proof. By definition, we have for a.e.z∈Ω
GhIsfz 2
v∈R κv
∇Isfz
, v z, v
2
v∈R κv 1
z, v
1
0
N
i1 ∂f ∂zi
tzvitsdt
1
0
2
v∈R κv
∇ftz, v tz, v t
s1dt
1
0
Ghftzts1dt
Is2Ghfz.
3.7
Similarly
DhIsfz −2
v∈R κv
Isf
z−Isf
σvz z, v2 |v|
2
−2
v∈R κv |v|
2
z, v2
1
0
ftz−ftσvz
ts−1dt
1
0 −2
v∈R κv |
v|2 tz, v2
ftz−ftσvz
ts1dt
Is2Dhfz.
It is also easy to see
ΔIsfz Is2Δfz. 3.9
Since Δh Δ Gh Dh, it follows that ΔhIsfz Is2Δhfz for a.e. z ∈ Ω. From
the regularity property of Dunkl operators, Δh maps C2Ωinto CΩ. By the continuity,
Lemma 3.2follows.
Lemma 3.3. LetH1{f ∈HΩ:Δhf0}. Ifs >0andQjas in1.12, then
RsH1H1, IsH1H1, QjH1H1. 3.10
Proof. Note that3.6implies
Rs2Δhfz ΔhRsfz, z∈Ω. 3.11
Indeed, from Lemma 3.1, Rs2Δh Rs2ΔhIsRs Rs2Is2ΔhRs ΔhRs. As direct
consequence of3.6and 3.11, we find that Isf and Rsf are Dunkl harmonic, whenever
fis Dunkl harmonic. From the definition ofQj, we thus obtainQjH1H1.
Lemma 3.4. Letg ∈HΩ,j∈N, Then for anyz∈Ω
Δh
z2jgxz2jΔhgx 4jz2j−1RN2j−2/2γ2jgz. 3.12
Proof. For anyf, g∈HΩ
ΔfgΔfg2∇f,∇gfΔg. 3.13
Takefz z2jand apply identities∂/∂z
iz2j 2jziz2j−1andΔz2j 2jN2j−2z2j−1
to yield
Δz2jgz2jΔg4jz2j−1RN2j−2/2g. 3.14
By our assumptionR ⊂RN. Therefore
vv, v∈R. 3.15
As a result,
for anyz∈Ωandv∈R. Indeed
σvz2 N
j1
zj−2
z, v |v|2 vj
2
z2−4z, vz, v
|v|2 4
z, v2v, v |v|4
z2.
3.17
Then
Dh
z2jgz−2
v∈R κv
z2jgz−σ
vz2jgσvz z, v2 |v|
2
z2jDhgz.
3.18
By definition, we have
Gh
z2jg2
v∈R κv
∇z2jg, v z, v
z2jG h
g2
v∈R κv
∇z2j, v z, v g
z2jGh
g4jγz2j−1gz.
3.19
SinceΔh Δ GhDh, summing up the above identity leads to identity3.12forz∈Ω.
Lemma 3.5. For any complex Dunkl harmonic functionfinΩ,
Δn
hz2nQnfz fz, z∈Ω. 3.20
Proof. From1.12andLemma 3.1, we know that
Qn−14nn!RN2n−1/2γRN2n−2/2γ· · ·RN/2γ. 3.21
DenotegQnf. Thengis Dunkl harmonic inΩdue to3.10, and
fz 4nn!RN2n−1/2γRN2n−2/2γ· · ·RN/2γgz. 3.22
We need to show
Δn
hz2ngz 4nn!RN2n−1/2γRN2n−2/2γ· · ·RN/2γgz 3.23
Letgbe Dunkl harmonic inΩandn∈N. ThenLemma 3.4shows
Δh
z2ngz4nz2n−1R
N2n−1/2γgz. 3.24
We use induction onnto prove3.23. It is easy to prove whenn 1. For the general case, from3.24we have
Δn h
z2ngz Δnh−1Δh
z2ngz
4nΔn−1
h
z2n−1R
N2n−1/2γgz
.
3.25
Equation3.23follows directly from the assumption of induction.
Now we come to the proof of our main theorem.
Proof ofTheorem 1.2.DenoteHn{f∈HΩ:Δhnf0}. It is sufficient to show that
HnHn−1Tn−1H1, n∈N, 3.26
whereTnz2nI. Notice thatLemma 3.5states that
Δn
hTnQnI. 3.27
We split the proof into two parts.
iHn⊃Hn−1Tn−1H1. SinceHn−1⊂Hn, we need only to showTn−1H1⊂Hn. For any
g ∈H1, by3.27and3.10we have
Δn h
Tn−1g
Δh
Δn−1
h Tn−1Qn−1
Qn−1−1g ΔhQn−1−1g0. 3.28
iiHn⊂Hn−1Tn−1H1. For anyf∈Hn, we have the decomposition
f I−Tn−1Qn−1Δnh−1
fTn−1
Qn−1Δnh−1f
. 3.29
We will show that the first summand above is inHn−1and the item in the braces of the second
summand is inH1. This can be verified directly. First,
Δn−1
h
I−Tn−1Qn−1Δn−1
fΔnh−1−Δhn−1Tn−1Qn−1
Δn−1
h
f
Δn−1
h −Δ n−1
h
f0.
3.30
Next, sinceΔnh−1f ∈H1andQn−1H1 ⊂H1, we haveQn−1Δnh−1f ∈H1, as desired.This proves
Next we prove that for anyf∈Hnthe decomposition
fgTn−1fn, g ∈Hn−1, fn∈H1 3.31
is unique. In fact, for such a decomposition, applyingΔnh−1on both sides we obtain
Δn−1
h f Δnh−1g Δhn−1Tn−1fn
Δn−1
h Tn−1Qn−1Q−
1
n−1f1
Qn−1−1fn.
3.32
Therefore
fnQn−1Δnh−1f, 3.33
so that
gf−Tn−1fn
I−Tn−1Qn−1Δn−1
f. 3.34
Thus the uniqueness follows by induction.
To prove the converse, we see from 3.23 that, for any n ∈ N, Δn1
h z2nH1 0.
Replacingnbyj, we have
Δn
hz2jH10 3.35
for anyn > j.
4. Gauss Decomposition
As a direct consequence of Theorem 1.2, we can get the extended Fischer decomposition theorem. LetPm denote the space of homogeneous polynomials of degreeminCN. Notice
thatDjmapsPmintoPm−1, so thatΔhPm⊂ Pm−2. Iff∈ Pm, then
Δm/2 1
h fz 0, 4.1
andIsfz 1/msfzso that
Qjfz dj,mfz, 4.2
whered−j,n1 4jj!N/2γnjandanaa1· · ·an−1. Denote
cjdj,m−2j 1
4jj!N/2γm−2j j
Corollary 4.1. Letf be a homogeneous polynomial of degreemin CN. Then there exist uniquely
Dunkl harmonic homogeneous polynomialsfjof degreem−2jsuch that
fz f0z z2f1z · · ·z2m/2fm/2z, ∀z∈Ω. 4.4
Moreover the Dunkl harmonic functionsf0, . . . , fm/2 are given by the following formulae:
f0
I−c1z2Δh
I−c2z4Δ2h
· · ·I−cm/2z2m/2Δhm/2
fz
f1c1Δh
I−c2z4Δ2h
· · ·I−cm/2z2m/2Δhm/2
fz
.. .
fm/2−1cm/2−1Δhm/2−1
I−cm/2z2m/2Δhm/2
fz
fm/2 cm/2Δhm/2fz.
4.5
Proof. Letf ∈ Pm, thenfis Dunkl harmonic of degreem/2 1, so thatTheorem 1.2gives
the decomposition offas in4.4. It remains to check the formulae off0, . . . , fm/2. We only
consider the formula off0, since the others are similar. That is, we need to show
f0
I−z2Q1Δh
I−z4Q2Δ2h
· · ·I−z2m/2Qm/2Δhm/2
fz
I−c1z2Δh
I−c2z4Δ2h· · ·I−cm/2z2m/2Δhm/2
fz.
4.6
Notice that for anyf∈ Pm,Δhm/2 f∈ Pm−2m/2 ∩H1, so that4.2implies
Qm/2Δhm/2 fz cm/2Δhm/2fz. 4.7
Therefore
z2m/2 Qm/2Δhm/2 fz cm/2z2m/2Δhm/2fz∈ Pm, 4.8
and also
I−z2m/2Qm/2 Δhm/2
fz I−cm/2 z2m/2 Δhm/2
fz∈ Pm. 4.9
The remaining proof follows by induction.
Acknowledgment
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