Integro-differential-difference equations associated with the Dunkl operator and entire functions

Integro-differential-difference equations associated with the Dunkl operator and entire functions

N´ejib Ben Salem, Samir Kallel

Abstract. In this work we consider the Dunkl operator on the complex plane, defined by Dkf(z) = d

dzf(z) +kf(z)−f(−z) z , k≥0.

We define a convolution product associated withDkdenoted∗kand we study the integro- differential-difference equations of the typeµ∗kf=^P∞_n=0a_n,kDⁿ_kf, where (an,k) is a sequence of complex numbers andµis a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.

Keywords: Dunkl operator, Fourier-Dunkl transform, entire function of exponential type, integro-differential-difference equation

Classification: 30D15, 33E30, 34K99, 44A35, 45J05

Introduction

In this paper we consider the first-order differential-difference operator onC Dkf(z) = d

dzf(z) +kf(z)−f(−z)

z , z∈C, f ∈ A(C)

(A(C) is the space of entire functions), which is known as the Dunkl operator of index k, k ≥ 0. It was introduced by C.F. Dunkl (see [4], [5]) and has found a wide area of applications in mathematics and mathematical physics.

It has been shown that there exists a unique intertwining operatorV_kbetween D_k andD= _dz^d which satisfies

V_kD=D_kV_k, V_kf(0) =f(0), for all f ∈ A(C).

By using the method of generalized Taylor series, we associate withD_kthe trans- lation operatorsT_z^k,z∈C, defined onA(C) by

(1) ∀ω∈C, T_z^kf(ω) =

∞

X

n=0

b_n(ω)Dⁿ_kf(z),

where bn(ω) = V_k(^ω_n!ⁿ). For an appropriate measure µover the real line and an entire function f, we define the convolution product ofµand f associated with D_k, denotedµ∗_kf and given by

∀z∈C, µ∗_kf(z) = Z

R

T_−y^k f(z)dµ(y).

In this work, we are interested in the study of the following integro-differential- difference equations

(2) µ∗kf(z) =X

n≥0

a_n,kDⁿ_kf(z),

where (a_n,k)_n≥0 is a sequence of complex numbers.

These equations characterize a class of entire functions of exponential type which intervenes in classical complex analysis and have many applications in other fields (for more details, one can see [3]). In fact this study shows that when the mea- sureµsatisfiesR

Re^σ|x|d|µ|<∞, whereσis a positive number, then every entire function of exponential type less thanσ, is a solution of such equations and con- versely iff is aC^∞-function onRsatisfying the equation (2) and ifP

n≥0a_n,kzⁿ is analytic inside the disk |z| ≤ a, a ≤ σ, then f is the restriction to R of an entire function of exponential type at mosta. After, we develop a method which permits us to construct solutions of these equations which are expressed in terms of normalized spherical Bessel functions of indexα

j_α(z) = Γ(α+ 1)

+∞

X

n=0

(−1)ⁿ n!

(^z₂)²ⁿ

Γ(n+α+ 1), z∈C.

Next, we suppose thatk >0. In this case, the restriction onRof the translation operators associated with the Dunkl operator given by formula (1) possess an integral representation which is available for a continuous function onR, so that we can consider equations of the type

(3) µ∗_kf =

N

X

n=0

a_n,kDⁿ_kf, a_N,k6= 0, N ∈N,

when f is a C^N-function on R and µ is an appropriate measure. We establish, under some assumptions, that every C^N-function on R satisfying equation (3) is a C^∞-function on R. In particular, if when all but one of the a_n,k are zero, 0 ≤ n ≤N, then f is the restriction on R of an entire function of exponential type.

We point out that the notion of integro-differential equations was analyzed in details by M.H. Mugler [9] in the classical case (which corresponds tok= 0).

Later, N. Ben Salem and W. Masmoudi have studied the integro-differential equa- tions associated with the Bessel differential operator (see [2]).

The paper is arranged as follows. The first section of this paper is devoted to the study and recall of some results of harmonic analysis associated with the Dunkl operatorD_k. Especially we define the translation operators and convolu- tion product associated withD_kof an appropriate measure and an entire function, we define also the Laplace-Dunkl transform of a measure over the real line and we establish some properties related with these objects.

In the second section, we deal with the integro-differential difference equations of the type µ∗_kf = P

n≥0a_n,kDⁿ_kf. We give a class of functions which are solutions of that equations.

In the third section, we assume that k > 0. We study equations of type µ∗_kf =PN

n=0a_n,kDⁿ_kf, wheref is aC^N-function onR.

In the last section we establish a Paley-Wiener type theorem associated with D_k and give some applications. For instance, we proceed to develop conditions on the measureµsuch equation of the formµ∗kf =PN

n=0a_n,kDⁿ_kf characterize the class of entire functions of exponential type a which are square integrable with respect to |x|^2kdx and bounded on the real line. Next, we continue by considering the equations characterizing entire functions of exponential type which have polynomial growth on the real line. The section closes by considering an equivalent condition characterizing the last equations in terms of the moments of the measureµ.

1. Harmonic analysis associated with the Dunkl operator We consider the following spaces:

- E(R) the space of C^∞-functions, endowed with the usual topology of uniform convergence of the functions and their derivatives of all order on compact subsets ofR;

-E^′(R) the space of distributions onRwith compact support;

-A(C) is the space of entire functions onCprovided with the topology of uniform convergence on every compact ofC;

-A^′(C) is the topological dual ofA(C);

- Exp(C) is the space of entire functions of exponential type, we have Exp(C) = [

a>0

Exp_a(C),

where

Exp_a(C) =

f ∈ A(C), Na(f) = sup

λ∈C

|f(λ)|e^−a|λ|<+∞

.

We provide Exp_a(C) with the topology defined by the norm Na(f). For this topology Exp_a(C) is a Banach space. Exp(C) is endowed with the inductive limit topology.

The Dunkl operatorDk associated with the parameterk≥0, is defined onC by

D_k(f)(z) = d

dzf(z) +kf(z)−f(−z)

z , f ∈ A(C).

Fork= 0, D₀ reduces to the usual derivative which will be denoted byD. It is well known that there exists a unique isomorphismV_k ofA(C) such that (4) V_kDf =D_kV_kf, V_kf(0) =f(0).

The operatorV_kis called the Dunkl intertwining operator of indexkbetweenD_k andD= _dz^d on the spaceA(C), (see [1]).

Fork >0,V_k has the following representation (see [5, Theorem 5.1]) (5) V_kf(z) =2^−2kΓ(2k+ 1)

Γ(k)Γ(k+ 1) Z ₁

−1

f(zt)(1−t²)^k−1(1 +t)dt, f∈ A(C).

Fork≥0, andλ, z ∈C, the equation

(D_ku(z) =λu(z), u(0) = 1, has a unique solutionφ^k_λ,0 given by

φ^k_λ,0(z) =j_k−¹

2(iλz) + λz 2k+ 1j_k+¹

2(iλz),

wherej_α is the normalized spherical Bessel function defined forα≥ −¹₂, by

j_α(z) = Γ(α+ 1)

+∞

X

n=0

(−1)ⁿ n!

(^z₂)²ⁿ

Γ(n+α+ 1), z∈C.

We remark thatφ^k_λ,0(z) =V_k(e^λ.)(z). Formula (5) and the last result imply that (6) |φ^k_λ,0(z)| ≤e^|λ||z|, |φ^k_λ,0(x)| ≤e^|x||Reλ|, |φ^k_−iy,0(x)| ≤1,

for allx, y∈Randλ, z∈C.

The function (λ, z)7−→φ^k_λ,0(z) (called Dunkl kernel) is analytic onC×C. There- fore, there exist unique analytic functionsb_n,n∈N, onCsuch that

φ^k_λ,0(z) =

∞

X

n=0

bn(z)λⁿ, λ, z∈C, wherebn(z) =V_k(ωⁿ

n!)(z) = 1 n!

dⁿ

dλⁿφ^k_λ,0(z)|_λ=0, namely

b_2n(z) = 1 (k+¹₂)nn!

z 2

2n

, b_2n+1(z) = 1 (k+¹₂)_n+1n!

z 2

2n+1

, ∀n∈N. We remark that for allz∈Cand for alln∈N, we have

(7) D_kb_n+1=bn and |bn(z)| ≤ |z|ⁿ n! . In the same context, we denote by

(8) φ^k_λ0,n(x) =V_k(xⁿe^λ0x) = dⁿ

dtⁿφ^k_t,0(x)_|t=λ0, n∈N and λ₀ ∈C.

Definition 1.1. Thetranslation operators associated with the Dunkl operator, denoted byT_z^k,z∈C, are defined onA(C) by

∀ω∈C, T_z^kf(ω) =

∞

X

n=0

bn(ω)Dⁿ_kf(z).

We next collect some properties of translation operators.

Proposition 1.2. The operatorsT_z^ksatisfy the following properties.

(i) For everyz∈C, the operatorT_z^k is linear and continuous map from A(C) into itself and

T_z^kf(ω) =V_k,zV_k,ωh

V_k⁻¹(f)(z+ω)i

, ω∈C.

(We use the notationV_k,zwhen we wish to emphasize the functional depen- dence on the variablez).

(ii) For all functionf inA(C)and for everyz∈C,z^′ ∈Cand ω∈C, we have T₀^k=identity, T_z^kf(ω) =T_ω^kf(z), D_kT_z^k=T_z^kD_k and T_z^kT_z^k′ =T_z^k′T_z^k. (iii) The function (z, ω) −→ T_z^kf(ω) is the unique solution of the following

Cauchy problem

(D_k,zu(z, ω) =D_k,ωu(z, ω), u(z,0) =f(z).

(iv) For allz∈C, ω∈Candλ∈C, we have

T_z^kφ^k_λ,0(ω) =φ^k_λ,0(z)φ^k_λ,0(ω) (product formula).

Remark. Fork >0, it was pointed out in [10], [11] that the translation operators T_x^k, x∈R, may be represented as:

(9) ∀y∈R, T_x^kf(y) = Z

R

f(z)dµ^k_x,y(z), f∈C(R),

(C(R) is the space of continuous functions onR),µ^k_x,yis a real bounded measure onRwith support in [−|x| − |y|,−||x| − |y||]∪[||x| − |y||,|x|+|y|], forx, y 6= 0, µ^k_x,y(R) = 1 andkµ^k_x,yk ≤4, for allx, y∈R.

Let us now recall the following generalized Taylor formula with integral re- mainder (see [8]), which will be used frequently.

Theorem 1.3. Letf be a function of class Cⁿ⁺¹ onR, n∈ N. Then we have the following generalized Taylor formula with integral remainder

f(x) =

n

X

p=0

bp(x)D^p_kf(0) + Z |x|

−|x|

Wn(x, y)D_kⁿ⁺¹f(y)|y|^2kdy,

where {Wn}, n = 0,1,2. . ., is a sequence of functions constructed inductively from the function|y|^2k and satisfying

Z |x|

−|x|

|Wn(x, y)||y|^2kdy≤b_n+1(|x|) +|x|bn(|x|).

Definition 1.4. (i) The Borel-Dunkl transform of an analytic functional S ∈ A^′(C) is defined by

Fk(S)(λ) =hS, φ^k_λ,0(.)i, λ∈C.

(ii) TheFourier-Dunkl transform of a distributionµin E^′(R) is defined by F_k(µ)(λ) =hµ , φ^k_−iλ,0(.)i.

(iii) Thek-convolution of two distributionsµ, ν∈ E^′(R) is given by hµ∗kν , fi=hµx, hνy, T_x^kf(y)ii, f ∈ E(R).

Next, let us recall the following Paley-Wiener type theorem associated with the operatorD_k (for some details see [1]).

Theorem 1.5. The Borel-Dunkl transformF_kis a topological isomorphism from A^′(C)ontoExp(C).

Theorem 1.6 (P´olya representation). If f is an entire function of exponential typea, a >0, thenf has the following integral representation

f(z) = 1 2iπ

Z

|ω|=a+ǫ

φ^k_z,0(ω)F(ω)dω,

whereǫ >0and F is an analytic function outside the disk centered at the origin and with radiusa.

Proof: From the Paley-Wiener Theorem 1.5, there exists an analytic functional S∈ A^′(C) such that

∀z∈C, f(z) =hS, φ^k_z,0(.)i.

Since the analytic functionalS is given by a complex measureµwith support in the disk centered at the origin and with radiusa, (see [7]), we have

∀z∈C, f(z) = Z

C

φ^k_z,0(ω)dµ(ω).

On the other hand, by using the Cauchy integral formula, we can write for all z∈C

φ^k_z,0(ω) = 1 2iπ

Z

|ξ|=a+ǫ

φ^k_z,0(ξ)

ξ−ω dξ, ǫ >0.

From Fubini’s Theorem we deduce that f(z) = 1

2iπ Z

|ξ|=a+ǫ

φ^k_z,0(ξ)F(ξ)dξ,

whereF(ξ) =R

C dµ(ω)

ξ−ω , (F is called the Borel Transform of the measureµ).

Proposition 1.7. Letf be an entire function of exponential typea,a >0. Then (i) for everyn∈N, the functionDⁿ_kf is entire and of exponential typea;

(ii) for everyz, ω∈Candǫ >0

|T_z^kf(ω)| ≤Cǫe(a+ǫ)(|z|+|ω|), whereC_ǫis a positive constant.

Proof: (i) It is clear that for n ∈N, the function Dⁿ_kf is entire. Let us show that Dⁿ_kf is of exponential type a. From the P´olya representation Theorem 1.6, we deduce

Dⁿ_kf(z) = 1 2iπ

Z

|ω|=a+ǫ

ωⁿφ^k_z,0(ω)F(ω)dω.

Now using the property of the functionφ^k_z,0, relation (6), we obtain (10) |D_kⁿf(z)| ≤(a+ǫ)ⁿ⁺¹M_ǫe^(a+ǫ)|z|,

whereM_ǫ= sup{|F(ω)|; |ω|=a+ǫ}. Sinceǫis arbitrary, we conclude thatD_kⁿf is of exponential typea.

(ii) We have forf ∈Exp_a(C),T_z^kf(ω) =P∞

n=0bn(ω)D_kⁿf(z). A combination of

(7) and (10) gives the result.

Notation. Forσ >0, letM_σ(R) be the space of Radon measures onRsatisfying Z

R

e^σ|x|d|µ|(x)<∞.

Definition 1.8. Let f be an entire function of exponential type a > 0 and µ ∈ Mσ(R) with σ > a. The convolution product associated with D_k of the functionf and the measureµis the function denotedµ∗_kf, defined by

∀z∈C, µ∗_kf(z) = Z

R

T_−y^k f(z)dµ(y).

Proposition 1.9. Let f be an entire function of exponential type a > 0 and µ∈M_σ(R)withσ > a. Thenµ∗_kf ∈Exp_a(C).

Proof: We have for allz∈C µ∗_kf(z) =

Z

R

T_−y^k f(z)dµ(y).

With the hypotheses on the measureµ, we deduce easily thatz7−→µ∗_kf(z) is entire. Now, letǫ∈R, 0< ǫ < σ−a, by using Proposition 1.7(ii) we have

∀z∈C, |µ∗_kf(z)| ≤C_ǫ^′e^(a+ǫ)|z|,

whereC_ǫ^′ is a positive constant. Sinceǫ is arbitrary, we deduce thatµ∗_kf is of

exponential typea.

2. Integro-differential-difference equations associated with the Dunkl operator for the class of entire functions of exponential type

Definition 2.1. Letµbe a measure inMσ(R). TheLaplace-Dunkl transform of the measureµis the function denotedLk(µ), defined by

L_k(µ)(z) = Z

R

φ^k_−z,0(y)dµ(y).

We remark that for µ ∈ M_σ(R), the function L_k(µ) is analytic in the strip

|Rez| ≤σ.

Theorem 2.2. Letµbe a measure inMσ(R). If the equation (11) ∀z∈C, µ∗_kf(z) =

∞

X

n=0

a_n,kDⁿ_kf(z), a_n,k∈C

is satisfied by any function f in Exp_a(C), 0 < a < σ, where P

n≥0a_n,kzⁿ is analytic in the closed disk centered at the origin and with radiusa, then

a_n,k= 1 n!

dⁿ

dzⁿL_k(µ)(z)_|z=0.

Conversely, if the sequence (a_n,k)_n≥0, is related to the measure in this fashion, then (11)holds for each class of entire function of exponential type a with 0 <

a < σ.

Proof: Let λ₀ ∈C such that|λ₀| ≤a. Since the function z 7−→φ^k_λ

0,0(z) is of exponential type|λ₀|, we have

µ∗_kφ^k_λ0,0(z) =φ^k_λ0,0(z) Z

R

φ^k_−λ0,0(y)dµ(y) =φ^k_λ0,0(z)L_k(µ)(λ₀).

On the other hand, we have X

n≥0

a_n,kD_kⁿφ^k_λ0,0(z) =φ^k_λ0,0(z)X

n≥0

a_n,kλⁿ₀.

So we deduce that

L_k(µ)(λ₀)−X

n≥0

a_n,kλⁿ₀

φ^k_λ0,0(z) = 0.

Takingz= 0, we obtainLk(µ)(λ0) =P

n≥0a_n,kλⁿ₀. This holds for everyλ₀such that|λ₀| ≤a.

Soa_n,k=_n!¹ _dz^dnnL_k(µ)(z)_|z=0. Conversely, letf ∈Exp_a(C), we have µ∗_kf(z) =

Z

R

X

n≥0

bn(−y)D_kⁿf(z)

dµ(y) =X

n≥0

Z

R

bn(−y)dµ(y)

Dⁿ_kf(z).

The last identity is justified by the fact that Z

R

X

n≥0

|bn(−y)||D_kⁿf(z)|d|µ|(y)<+∞

which is a consequence of Proposition 1.7(i) and the relation (7). We conclude by observing that

Z

R

bn(−y)dµ(y) = 1 n!

Z

R

dⁿ

dλⁿφ^k_−λ,0(y)_|λ=0dµ(y) = 1 n!

dⁿ

dλⁿL_k(µ)(λ)_|λ=0.

Theorem 2.3. Letµ∈Mσ(R)and letf be aC^∞-function onRsatisfying

∀x∈R, µ∗_kf(x) =X

n≥0

a_n,kDⁿ_kf(x),

where(a_n,k)_n≥0 is a sequence of complex numbers such that the series P

n≥0a_n,kzⁿ is analytic inside the closed disk|z| ≤a,0< a < σ. Then f is the restriction onRof an entire function of exponential type at mosta.

Proof:Letxbe fixed inR. From the convergence of the seriesP

n≥0a_n,kDⁿ_kf(x), we deduce that there existsN₁(x)∈N such that|a_n,kDⁿ_kf(x)| ≤1, for all n ≥ N₁(x). On the other hand, since the seriesP

n≥0a_n,kzⁿ is analytic in the disk

|z| ≤ a, we have lim sup_n−→+∞|a_n,k|¹ⁿ ≤ ¹_a. Thus for every ǫ,0 < ǫ < _a¹, there exists N₂ ∈ N such that, |a_n,k|¹ⁿ > (¹_a −ǫ), for all n ≥ N₂. Hence, for n≥max (N₁(x), N₂), we have|Dⁿ_kf(x)| ≤(_a¹ −ǫ)⁻ⁿ. By applying the Delsarte Taylor formula with integral remainder given in Theorem 1.3, to the functionf and the relation (7), we obtain

f(x) =

N

X

n=0

bn(x)Dⁿ_kf(0) +R_N(x), where

|RN(x)| ≤ sup

0≤|t|≤|x|

|D_k^N+1f(t)|(b_N+1(|x|) +|x|bN(|x|))

≤ |x|^N+1

(N+ 1)!(2 +N) sup

0≤|t|≤|x|

|D^N+1_k f(t)|.

For eacht, 0≤ |t| ≤ |x|, the above analysis shows that there exists N_t∈Nsuch that

|D_kⁿf(t)| ≤(1

a−ǫ)⁻ⁿ, for n≥N_t.

But [−|x|,|x|] is compact, so there is someN^′ independent oft such that sup

0≤|t|≤|x|

|D_kⁿf(t)| ≤(1

a−ǫ)⁻ⁿ, for n≥N^′.

Hence forN ≥N^′, we have|R_N(x)| ≤ _(N^|x|N+1_+1)!(2+N)(_a¹−ǫ)^−(N+1), which implies that R_N(x) tends to zero asN tends to +∞. Consequently the functionf may be expanded in a generalized Taylor seriesf(x) =P

n≥0bn(x)Dⁿ_kf(0). Hencef is the restriction onRof the entire function ggiven byg(z) =P

n≥0bn(z)Dⁿ_kf(0).

We deduce from (7) and (10) thatgis of exponential type at mosta.

Example. Letµbe the measure defined by dµ(y) = a

2e^−a|y||y|^2kdy, a >0, k≥0.

Consider the equation

(12) µ∗_kf =

∞

X

n=0

a_n,kD²ⁿ_k f,

wherea_n,k is given by

a_n,k= 2^2kΓ(k+¹₂)Γ(k+n+ 1) n!Γ(¹₂)a^2(k+n) . By computation, we have

L_k(µ)(z) = 2^2ka²Γ(k+¹₂)Γ(k+ 1) Γ(¹₂)(a²−z²)^k+1 =

∞

X

n=0

a_n,kz²ⁿ, for |z|< a.

Due to Theorem 2.2, Theorem 2.3 shows that equation (12) characterizes entire functions of exponential type less thana.

This example shows that the relation between exponential type and domain of analyticity is sharp sinceL_k(µ) has a singularity at ±a.

We proceed now to develop a method which permits us to construct solutions of (11), expressed in terms of functions given by (8).

Proposition 2.4. Letµ∈M_σ(R)andλ₀ a zero of multiplicityN of the function g(z) =

s

X

n=0

c_n,kzⁿ− L_k(µ)(z),

where (c_n,k)_0≤n≤s is a finite sequence in C. The function defined by f(x) = PN−1

m=0amφ^k_λ

0,m(x)is a solution of the equationµ∗kf(x) =Ps

n=0c_n,kDⁿ_kf(x), where(am)_{0≤m≤N−1} is a finite sequence inC.

Proof: We have, for allx∈R µ∗_kf(x) =

N−1

X

m=0

am m

X

j=0

m j

(L_k(µ))^(j)(λ₀)φ^k_λ0,m−j(x).

Now we use the fact thatλ₀ is a zero of multiplicityN of the function g, so we have

(L_k(µ))^(j)(λ₀) = ( Ps

n=jj! ⁿ_j

c_n,kλ^n−j₀ , for 0≤j≤s,

0, if s < j ≤N.

Hence

µ∗_kf(x) =

s

X

n=0

c_n,kV_k,y





n

X

j=0 N−1

X

m=j

am

m j

j!

n j

y^m−je^λ0yλ^n−j₀



(x).

Using the following relation which is obtained by the generalized product rule, Dⁿ

(_N−1 X

m=0

amx^me^λ0x )

=e^λ0x

n

X

j=0 N−1

X

m=j

am

m j

j!

n j

x^m−jλ^n−j₀

and (4), we deduce the result.

In [1], we have called the functions φ^k_λ,m k-exponential-monomials which can be expressed in terms of normalized spherical Bessel functions, namely

φ^k_λ,m(x) =





 x²ⁿPn

s=0f_2n,sφ^k+s_λ,0 (x), ifm= 2n,

x²ⁿ⁺¹Pn s=0f_2n,s

φ^k+s_λ,0 (x)− ^k+s

k+s+¹₂j_k+s+1 2(iλx)

, ifm= 2n+ 1, wheref_2n,s are given by

f_2n,s= (−1)^s n

s

(k)s

(k+¹₂)s

, 0≤s≤n.

We can extend the previous proposition to infinite case.

Proposition 2.5. Letµ∈Mσ(R)andλ₀ a zero of multiplicityN of the function g(z) =

∞

X

n=0

c_n,kzⁿ− L_k(µ)(z).

Suppose that the series P

n≥0c_n,kzⁿ is analytic in the disk |z| ≤ a < σ and

|λ₀|< a. Then every function of the formf(x) =PN−1 m=0amφ^k_λ

0,m(x)is a solution of the integro-differential-difference equation

µ∗kf =

∞

X

n=0

c_n,kD_kⁿf.

Proof: We proceed as in the previous proof, we remark that we can change the order of summation by using the uniform convergence of series.

Lemma 2.6. If ψis analytic on a neighborhood of a contourγin Cand Z

γ

φ^k_z,0(ω)ψ(ω)dω= 0, thenψis analytic insideγ.

Proof: It is obtained in the same way as for Lemma 6.10.6, p. 110 in [3].

In the following proposition, we show that every solution of equation (11) which is entire of exponential type is a sum ofk-exponential-monomials functions.

Proposition 2.7. Let f be an entire function of exponential type a, a > 0, P

n≥0a_n,kzⁿ an analytic function in a closed disk |z| ≤ b, which contains the conjugate indicator diagram of f and µ ∈ Mσ(R), with σ ≥ b. If moreover f satisfies the equation

µ∗_kf =

∞

X

n=0

a_n,kD_kⁿf.

Thenf is of the following form

f(z) =

m

X

s=0 ls−1

X

n=0

βn,sφ^k_λs,n(z), βn,s∈C,

where λ_s, 0 ≤ s ≤ m, are the zeros of multiplicity l_s of the function g(z) = L_k(µ)(z)−P∞

n=0a_n,kzⁿ, which are contained in the conjugate indicator diagram of f,(m is possibly infinite).

Proof: From P´olya representation Theorem 1.6, we have f(z) = 1

2iπ Z

|ω|=a+ǫ

φ^k_z,0(ω)F(ω)dω, z∈C, whereF is analytic outside the disk|z| ≤aandǫ >0. Hence we have

∀z∈C, µ∗_kf(z) = 1 2iπ

Z

|ω|=a+ǫ

φ^k_z,0(ω)L_k(µ)(ω)F(ω)dω.

On the other hand, we have

∞

X

n=0

a_n,kDⁿ_kf(z) = 1 2iπ

Z

|ω|=a+ǫ

∞

X

n=0

a_n,kωⁿ

!

φ^k_z,0(ω)F(ω)dω.

Sincef is a solution of the equation (11), we must have Z

|ω|=a+ǫ

φ^k_z,0(ω)

" _∞ X

n=0

a_n,kωⁿ− L_k(µ)(ω)

#

F(ω)dω= 0.

By using Lemma 2.6, we deduce that the function ω7−→

"_∞ X

n=0

a_n,kωⁿ− L_k(µ)(ω)

# F(ω)

is analytic inside the disk|z| ≤ a+ǫ. Hence the function F has at most poles at the zeros of the functionω7−→g(ω) =L_k(µ)(ω)−P_∞

n=0a_n,kωⁿ, contained in the conjugate indicator diagram off. Now by using the P´olya representation and the residue theorem, we deduce that

f(z) =

m

X

s=0

Res(φ^k_z,0(ω)F(ω), λs) =

m

X

s=0 ls−1

X

n=0

β_n,sφ^k_λ

s,n(z), whereβn,s =_(l ¹

s−1)!

ls−1 n

_dls−1−n

dω^ls−1−n

h(ω−λs)^lsF(ω)i

|ω=λs

.

3. Integro-differential equations associated with the Dunkl operator on the space of Cⁿ-functions on R

In the following we suppose thatk >0. Then the translation operatorsT_x^k, x∈ Rassociated with the Dunkl operator, are given for a continuous function onR by formula (9).

Definition 3.1. Letf be a continuous function on R. We say that the nonneg- ative functionψ∈C(R) is abounding function off, if we have

(i) ∀x∈R,|f(x)| ≤ψ(x),

(ii) there exists a constantA=A(ψ, k) such that

∀x, y∈R, Z

R

ψ(z)d|µ^k_x,y|(z)≤Aψ(x)ψ(y).

The smallest constant satisfying the latter inequality will be called the supporting constant.

Example. ψ(x) =e^a|x|,a >0, we have

∀x, y∈R Z

R

e^a|z|d|µ^k_x,y|(z)≤4e^a(|x|+|y|) which can be seen by using the properties of the measureµ^k_x,y.

Lemma 3.2. Letf be a function of classC^m onR,m∈N, such thatD_k^mf is of classCⁿonR,n∈N. Thenf is of class C^m+n onR.

Proof: See [8].

Lemma 3.3. Let f ∈ C¹(R) and µ be a measure on R. If ψ is a bounding function off andDkf, satisfying R

Rψ(−y)d|µ|(y)<+∞, thenµ∗kf ∈C¹(R) and we have

D_k(µ∗_kf) =µ∗_kD_kf.

Proof: Using the bounding function, by differentiation under the integral, we can see thatµ∗kf ∈C¹(R). On the other hand, we have by Theorem 1.3

D_k(µ∗_kf)(x) = lim

a−→0

T_x^k(µ∗_kf)(a)−µ∗_kf(x) b₁(a)

= lim

a−→0

Z

R

T_x^k(T_−y^k f)(a)−T_−y^k f(x) b₁(a) dµ(y), whereT_x^k(T_−y^k f)(a)−T_−y^k f(x) =R|a|

−|a|W₀(a, t)T_x^kT_−y^k (D_kf)(t)|t|^2kdt. Since

Z |a|

−|a|

W₀(a, t)T_x^kT_−y^k (D_kf)(t)|t|^2kdt

≤A² sup

|t|≤|a|

ψ(t)ψ(−y)ψ(x)(b₁(|a|) +|a|),

we have for 0< a <1

T_x^k(T_−y^k f)(a)−T_−y^k f(x) b₁(a)

≤A²

2Γ(k+³₂) Γ(k+¹₂) + 1

sup

|t|≤1

ψ(t)ψ(−y)ψ(x).

Asy7−→ψ(−y)∈L¹(d|µ|), the dominated convergence theorem and the following formula

a7−→0lim

T_a^k(T_−y^k f)(x)−T_−y^k f(x)

b₁(a) =D_k(T_−y^k f)(x) =T_−y^k (D_kf)(x)

yieldD_k(µ∗_kf)(x) = (µ∗_kD_kf)(x).

Lemma 3.4. Letf be in C^∞(R). Suppose that there exist a positive constant B and a nonnegative continuous functionψonRsuch that

∀x∈R, ∀n∈N, |Dⁿ_kf(x)| ≤Bⁿψ(x).

Thenf is the restriction toRof an entire function of exponential typeB.

Proof: From Theorem 1.3, we have for allx∈R, f(x) =

n−1

X

s=0

bs(x)D_k^sf(0) +Rn(x)

with

|Rn(x)| ≤ sup

|t|≤|x|

|Dⁿ_kf(t)|(bn(|x|) +|x|b_n−1(|x|))≤Bⁿ sup

|t|≤|x|

ψ(t)n+ 1 n! |x|ⁿ. The latter term goes at zero asn tends to +∞. Hence for x∈R, f(x) can be expanded as

f(x) =

∞

X

n=0

bn(x)D_kⁿf(0), x∈R. Put g(z) = P∞

n=0bn(z)Dⁿ_kf(0). This series defines an entire function and we have,|g(z)| ≤ψ(0)e^B|z|. Theng is entire of exponential type at mostB.

Proposition 3.5. Letf ∈C(R), ψ a bounding function off andµ a measure

onRsuch that Z

R

ψ(−t)d|µ|(t) =M <+∞.

If f is a solution of the equation

Dkf =µ∗kf,

thenf is the restriction toRof an entire function of exponential typeAM, where Ais the supporting constant.

Proof: Fixa, and choose a sequence such that xn −→a as n−→ +∞. Since forδ >0

|T_−y^k f(xn)−T_−y^k f(a)| ≤2A max

|x−a|≤δψ(x)ψ(−y)

and the latter terms is inL¹(d|µ|), the dominated convergence theorem yields

n−→+∞lim (D_kf(xn)− D_kf(a)) = 0.

Hence D_kf ∈C(R), so that Lemma 3.2 implies f ∈C¹(R). From the following inequality

|D_kf(x)| ≤ Z

R

|T_−y^k f(x)|d|µ|(y)≤AM ψ(x),

we see thatD_kf has a bounding functionAM ψ. By Lemma 3.3 we obtain D²_kf(x) =

Z

R

T_−y^k (D_kf)(x)dµ(y), so |D²_kf(x)| ≤(AM)²ψ(x).

An induction argument shows thatf ∈C^∞(R) and we have forn≥1 Dⁿ_kf(x) =µ∗_kD_kⁿ⁻¹f(x), with |Dⁿ_kf(x)| ≤(AM)ⁿψ(x), for n≥0.

The result follows from Lemma 3.4.

Example. Letµbe the measure defined by µ=X

s∈Z

τ 4(−1)^s

π²(2s+ 1)²δ₋2s+1 2τ π

whereδ_adenotes the Dirac point mass measure ata. Iff is bounded on the real axis and satisfies the equation

D_kf(x) =µ∗_kf(x) = 4τ π²

+∞

X

n=−∞

(−1)ⁿ (2n+ 1)²T^k2n+1

2τ πf(x),

thenf is the restriction onRof an entire function of exponential type|µ|(R) =τ.

Lemma 3.6. Letf ∈Cⁿ(R),n≥1, satisfying

∀x∈R, |Dⁿ_kf(x)| ≤Ae^τ|x|, whereA=A_k is a positive constant andτ >0. Then we have

|D^n−s_k f(x)| ≤τ^−s

2^sA+ 2^s−1τ C_n−1+ 2^s−2τ²C_n−2+· · ·+τ^sC_n−s e^τ|x|, whereC_n−s=|D^n−s_k f(0)|,0< s≤n.

Proof: From the Delsarte Taylor formula with integral remainder, Theorem 1.3, we deduce that

|Dⁿ⁻¹_k f(x)| ≤ |Dⁿ⁻¹_k f(0)|+ 2A Z _|x|

0

e^{τ y}dy

≤ 1

τ(2A+τ C_n−1)e^τ|x|.

We complete by induction.

From Proposition 2.7, we deduce that an entire function of exponential type which satisfies an equation of the form in the following proposition is a sum of k-exponential-monomials functions. The following proposition makes it clear why this hypothesis on the analyticity of the function is chosen, since a solution which is of exponential growth on the real line is shown to be entire of exponential type.

Proposition 3.7. Letf ∈Cⁿ(R)satisfy (i) |f(x)| ≤M e^τ|x|,

(ii) D_kⁿf =µ∗_kf, forn≥2,

where µis a measure onR such thatB =R

Re^τ|t|d|µ|(t)<+∞. Thenf is the restriction toRof an entire function of exponential type at most(4B)ⁿ¹.

Proof: We have

|Dⁿ_kf(x)| ≤ Z

R

|T_−t^k f(x)|d|µ|(t)≤4M Be^τ|x|. By Lemma 3.6, we deduce that

|D_k^n−sf(x)|

≤τ^−s

2^s×4M B+ 2^s−1τ C_n−1+ 2^s−2τ²C_n−2+· · ·+τ^sC_n−s e^τ|x|, for 0< s≤n. On the other hand, by using Lemma 3.3, we obtain

∀x∈R, ∀s∈N, 0≤s≤n, D^n+s_k f(x) =µ∗kD^s_kf(x).

So that,|D_k^n+sf(x)| ≤4BC˜_se^τ|x|, for 0< s < n, and|D_k²ⁿf(x)| ≤(4B)²M e^τ|x|, where ˜Cs = τ^s−n 2^n−s×4M B+ 2^n−s−1τ C_n−1+· · ·+τ^n−sCs

. Repeating this process we obtain form∈Nand 0< s < n,

|D_k^nm+sf(x)| ≤(4B)^mC˜_se^τ|x| and |D_k^nmf(x)| ≤(4B)^mM e^τ|x|. Hence we deduce by Lemma 3.3 thatf ∈C^∞(R). Furthermore, we have

lim sup

j−→+∞|D^j_kf(x)|

1

j ≤ lim

m−→+∞

(4B)^mC˜_snm+s¹

= (4B)ⁿ¹

for 0< s < nfixed. It then follows from Lemma 3.4 thatf is entire of exponential

type at most (4B)ⁿ¹.

Lemma 3.8. Letf ∈C^N(R)satisfy (i) |f(x)| ≤M e^τ|x|,

(ii) |PN

n=0a_n,kDⁿ_kf(x)| ≤Be^τ|x|,a_N,k6= 0.

Then|D^s_kf(x)| ≤M_s,ke^τ|x|,0< s≤N, whereM_s,kis a constant depending upon the function,k ands.

Proof: Proceed by induction onN. If|a_1,kDkf(x) +a_0,kf(x)| ≤Be^τ|x|, then

|D_kf(x)| ≤ 1

|a_1,k|(B+|a_0,k|M)e^τ|x|.

This yields the statement forN = 1. Now if|PN+1

n=0 a_n,kDⁿ_kf(x)| ≤Be^τ|x|, then the generalized Taylor formula with integral remainder gives

N

X

n=0

a_n+1,kD_kⁿf(x)

≤

N

X

n=0

a_n+1,kDⁿ_kf(0)

+ Z |x|

−|x|

|W0(x, y)||

N+1

X

n=1

a_n,kD_kⁿf(y)||y|^2kdy

≤h

P₀+ (|a_0,k|M+B)2τ⁻¹i e^τ|x|, where P₀ = |PN

n=0a_n+1,kDⁿ_kf(0)|. By our induction hypothesis, |D_k^sf(x)| ≤ M_s,ke^τ|x|, for 0< s≤N. Further

|D^N+1_k f(x)| ≤ 1

|aN+1,k|

" _N X

n=0

|a_n,k|M_n,k+B

# e^τ|x|.

This concludes the proof of Lemma 3.8.

Proposition 3.9. Letµbe a measure onRsuch thatB=R

Re^τ|t|d|µ|(t)<+∞, τ >0, andf ∈C^N(R)satisfy

(i) |f(x)| ≤M e^τ|x|,x∈R, (ii) µ∗_kf(x) =PN

n=0a_n,kDⁿ_kf(x),a_N,k6= 0.

Thenf is infinitely differentiable.

Proof: First

N

X

n=0

a_n,kDⁿ_kf(x)

≤4M Be^τ|x|.

Observe that by Lemma 3.8, |D^s_kf(x)| ≤ M_s,ke^τ|x|, 0 < s ≤ N. Further, Lemma 3.3 implies that the right hand side of the following equation

D_k^Nf(x) = 1

|a_N,k|

"

µ∗_kf(x)−

N−1

X

n=0

a_n,kD_kⁿf(x)

# , is differentiable and that in fact

D_k^N+1f(x) = 1

|a_N,k|

"

µ∗_kD_kf(x)−

N−1

X

n=0

a_n,kDⁿ⁺¹_k f(x)

# . Therefore

|D_k^N+1f(x)| ≤ 1

|a_N,k| ^N

X

n=1

|a_n−1,k|M_n,k+ 4M_1,kB

e^τ|x|.

This process can be repeated infinitely, proving the proposition.

Remark. Proposition 3.7 and 3.9 have shown that the equations µ∗_kf(x) = PN

n=0a_n,kD_kⁿf(x) have solutions which are entire functions of exponential type when all but one of the a_n,k are zero, 1 ≤ n ≤ N, and are at least infinitely differentiable otherwise.

4. Characterizations for certain classes of entire functions of slow growth on the real axis associated with the Dunkl operator

In this section we show a Paley-Wiener type Theorem associated withDkand we proceed to develop conditions on the measure (more precisely on the Fourier- Dunkl transform of the measure which will be defined below) such that equations of the form µ∗_kf = PN

n=0a_n,kDⁿ_kf characterize the class of entire functions of exponential type a which are square integrable with respect to |x|^2kdx and bounded on the real line.

Next, we continue by considering the equations characterizing entire functions of exponential type which have polynomial growth on the real line. The section closes by giving some other results by considering these same equations in the same order.

Notations. We denote by

• L^p_k(R), 1≤p <∞, the space of measurable functionsf onRsuch that kfk_p,k=

Z

R

|f(x)|^p|x|^2kdx ¹_p

<+∞.

• L²_k([−a , a]) the subspace of functions inL²_k(R) vanishing outside [−a , a], a >

0.

• Exp^B_a(C), the space of entire functions of exponential typeawhich are bounded on the real line.

• L²_k,a(R) the subspace of Exp^B_a(C) consisting of functions belonging to L²_k(R).

-The Fourier-Dunkl transform onL¹_k(R) is defined by F_k(f)(ξ) =c_k

Z

R

f(x)φ^k_−iξ,0(x)|x|^2kdx, wherec_k= ¹

2^{k+ 12}Γ(k+¹₂).

-Letµbe a finite Radon measure on the real line. The Fourier-Dunkl transform ofµis given by

F_k(µ)(y) = Z

R

φ^k_−iy,0(x)dµ(x).

Many properties of the Euclidean Fourier transform carry over to Fourier-Dunkl transform. In particular F_k(f)∈ C₀(R) for f ∈ L¹_k(R) (C₀(R) is the space of

continuous functions onR such that vanish at infinity), and there holds anL¹- inversion Theorem: If f,F_k(f) ∈ L¹_k(R) then f = F_kF_k(f) = F_kF_k(f) a.e, where Fk(f)(ξ) = Fk(f)(−ξ). Moreover, the Fourier-Dunkl transform Fk is a topological isomorphism from S(R) onto itself (S(R) is the Schwartz space of rapidly decreasing functions on the R), so Fk can be extended to a Plancherel transform onL²_k(R). For details see [6].

Letf in L^p_k(R), 1≤p <∞. We define the distributionS_f by (13) hS_f, ϕi_k=

Z

R

f(x)ϕ(x)|x|^2kdx, ϕ∈S(R).

Letf in A(C) be such that f(z) =

Z

R

g(t)φ^k_iz,0(t)|t|^2kdt, z∈C,

withg∈L²_k([−a , a]), a >0. Thenf is an entire function of exponential typea.

The following Paley-Wiener type theorem associated with the operatorD_kasserts the converse of this is true, if we know thatf restricted to the real axis belongs toL²_k(R). More precisely, we have

Theorem 4.1. Supposef ∈L²_k(R)∩ A(C). Then f(z) =c_k

Z a

−a

g(t)φ^k_iz,0(t)|t|^2kdt, whereg∈L²_k([−a , a])if and only if f is of exponential typea.

Proof: Suppose f is of exponential type a and its restriction to the real axis belongs toL²_k(R). Let gbe the Fourier-Dunkl transform off. Then

f(x) = lim

T−→+∞c_k Z T

−T

g(t)φ^k_ix,0(t)|t|^2kdt,

where the limit is in the topology ofL²_k(R). If t <−a, let Γ be the closed curve in the upper half plane which consists of the segmentL₁ = [−T,−ǫ], γǫ, ǫ > 0 is the small semicircle from−ǫto ǫ, oriented counterclockwise,L₂= [ǫ, T], L₃ = [T, T +iT], L₄ = [T +iT,−T +iT] and L₅ = [−T +iT,−T]. We can use a similar argument with Γ in the lower half plane ift > a. We obtain the result by proceeding in the same way as [3, Theorem 6.8.1, p. 103] and using that

ǫ−→0lim Z

γǫ

f(z)φ^k_ix,0(z)|z|^2kdz= 0.

Remarks. (i) Iff ∈L²_k,a(R), then for alln∈N,Dⁿ_kf ∈L²_k,a(R).

(ii) Letξ_k be the function defined by ξ_k(x) = Fk(χ_[−a,a]), where χ_[−a,a] is the characteristic function of the interval [−a, a]. We have

(14) ξ_k(x) = 1 2^k−21Γ(¹₂)

a^2k+1

∞

X

n=0

(−1)ⁿ(xa)²ⁿ (2n)!(2n+ 2k+ 1)

Γ(n+¹₂) Γ(n+k+¹₂).

The functionξ_k belongs toL²_k,a(R) and its Fourier-Dunkl transform equals 1 on the interval [−a, a]. In the casek= 0, we haveξ₀(x) =q

π2 sinax x .

Theorem 4.2. Letf ∈L²_k(R)∩C^N(R)andµ be a finite Radon measure onR such that

F_k(µ)(t) = ( PN

n=0a_n,k(it)ⁿ, for |t| ≤a and a_n,k are complex, g(t), for |t|> a where g(t)6=PN

n=0a_n,k(it)ⁿ. Then

(15) µ∗_kf(x) =

N

X

n=0

a_n,kDⁿ_kf(x)

if and only iff ∈L²_k,a(R). Further, if (15)holds for everyf ∈L²_k,a(R) but not for everyf ∈L²_k,b(R), whereb > a, thenF_k(µ)has the form above.

Proof: Supposef ∈L²_k,a(R). From the Paley-Wiener type Theorem 4.1 and the assumptions on the measureµ, we have

F_k(f)(t)

"

F_k(µ)(t)−

N

X

n=0

a_n,k(it)ⁿ

#

= 0.

Then

F_k(µ∗_kf)(t) =F_k(

N

X

n=0

a_n,kDⁿ_kf)(t).

So, we see that (15) holds. Conversely, if (15) holds, by the Fourier-Dunkl trans- form and the assumptions on the measure µ, it is clear that F_k(f)(t) = 0, for

|t|> a. Taking the inverse Fourier-Dunkl transform, we have

f(x) =c_k Z a

−a

φ^k_ix,0(t)F_k(f)(t)|t|^2kdt.