International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 2 (2015), 87-92
http://www.etamaths.com

A REAL PALEY-WIENER THEOREM FOR THE GENERALIZED
DUNKL TRANSFORM

A. ABOUELAZ, A. ACHAK , R. DAHER, EL. LOUALID∗

Abstract. In this article, we prove a real Paley-Wiener theorem for the generalized
Dunkl transform on R.

1. Introduction

In [3] N.B Andersen proved a real Paley-Wiener theorem for the dunkl transform. In this
paper, we first prove a real Paley-Wiener theorem for the generalized dunkl transform.
Let Λα denote the Dunkl operator and Fα,n the Dunkl transform, Chettaoui, C., Trimèche
proved in [4] the following theorem:

Theorem 1.1. Let 1 ≤ p ≤∞. Let f ∈S(R) (the Schwartz space on R). Then

lim
m→∞

‖Λmα f‖
1
m
p = sup{|λ|,λ ∈ SuppFα(f)}.

N.B Andersen in [3] gave a simple proof of the above theorem by using the real Paley-
Wiener theorem for the Dunkl transform. Our second result is to prove the above theorem
for the generalized Dunkl transform.
The structure of the paper is as follows: In section 2 we set some notations and collect
some basic results about the Dunkl operator and the Dunkl transform, and we give also
some facts about harmonic analysis related to the first-order singular differential-difference
operator Λα,n, and the generalized Dunkl transform. In section 3 we state and prove a
real Paley-Wiener theorem for the generalized Dunkl transform. In section 4 we give a
characterization of the support of the generalized Dunkl transform on R

2. Preliminaries

Throughout this paper we assume that α > −1
2
, and we denote by

• E(R) the space of functions C∞ on R, provided with the topology of compact con-
vergence for all derivatives. That is the topology defined by semi-norms

Pa,m(f) = supx∈[−a,a]

m∑
k=0

|
dk

dxk
f(x) |, a > 0, m = 0, 1, ...

• Da(R), the space of C∞ function on R, which are supported in [−a,a], equipped
with the topology induced by E(R).

• D(R) =
⋃
a>0 Da(R), endowed with inductive limit topology.

• En(R) (resp Dn(R)) stand for the subspace of E(R) (resp D(R)) consisting of func-
tions f such that

f(0) = ... = f(2n−1)(0).

2010 Mathematics Subject Classification. 65R10.
Key words and phrases. Real Paley-Wiener theorem; Generalized Dunkl transform.

c©2015 Authors retain the copyrights of their papers,
and all open access articles are distributed under the terms of the Creative Commons Attribution License.

87



88 OZTURK

• Lpα the class of measurable functions f on R for which ‖f‖p,α < ∞, where

‖f‖p,α =
(∫

R
|f(x)|p|x|2α+1dx

)1
p

, ifp < ∞,

and ‖f‖∞,α = ‖f‖∞ = esssupx≥0|f(x)|.
• Lpα,n the class of measurable functions f on R for which

‖f‖p,α,n = ‖M−1f‖p,α+2n < ∞.

• Dpα,n(R) = Dn(R)
⋂
Lpα,n(R).

• Ha, a > 0, the space of entire rapidly decreasing functions of exponential type a ;
that is, f ∈ Ha, a > 0 if and only if, f is entire on C and for all j=0,1......

qj(f) = sup
λ∈C
|(1 + λ)mf(λ)e−a|Imλ| < ∞

Ha, a > 0 is equipped with the topology defined by the semi-norms qj, j = 0, 1.....
• H =

⋃
a>0 Ha, equipped with the inductive limit topology.

2.1. Dunkl transform. In this subsection we recall some facts about harmonic analysis
related to Dunkl operator Λα associated with reflection group Z2 on R. We cite here, as
briefly as possible, only some properties. For more details we refer to [2, 4, 5].
The Dunkl operator Λα is defined as follow:

(1) Λαf(x) = f′(x) + (α +
1

2
)
f(x) −f(−x)

x
.

The Dunkl kernel eα is defined by

(2) eα(z) = jα(iz) +
z

2(α + 1)
jα+1(z) (z ∈ C)

where

jα(z) = Γ(α + 1)Σ
∞
n=0

(−1)n(z
2
)2n

n! Γ(n + α + 1)
(z ∈ C).

is the normalized spherical Bessel function of index α. The functions eα(λ) λ ∈ C, are
solutions of the differential-difference equation

Λαu = λu, u(0) = 1.

Furthermore, Dunkl kernel eα possesses the Laplace type integral representation

eα(z) = aα

∫ 1
−1

(1 − t2)α−
1
2 (1 + t)eztdt,

where

(3) aα =
Γ(α + 1)
√
πΓ(α + 1

2

).

The Dunkl transform of a function f ∈ D(R) is defined by

(4) Fα(f)(λ) =
∫
R
f(x)eα(−iλx)|x|2α+1dx, λ ∈ C.

Theorem 2.1. (i): The Dunkl transform Fα is a topological automorphism from
D(R) onto H. More precisely f ∈ Da(R) if, and only if, Fα(f) ∈ Ha



A REAL PALEY-WIENER THEOREM 89

(ii): For every f ∈ D(R) ,

f(x) =

∫
R
Fα(f)(λ)eα(iλx)|λ|2α+1dλ,∫

R
|f(x)|2|x|2α+1dx = mα

∫
R
|Fα(f)(λ)|2|λ|2α+1dλ,

where

(5) mα =
1

22(α+1)(Γ(α + 1))2
.

2.2. Generalized Dunkl transform. In this section, we recall some properties about
Generalized Dunkl transform. We refer to [1] for more details and references.
The first-order singular differential-difference operator on R is defined as follow

(6) Λα,nf(x) = f′(x) + (α +
1

2
)
f(x) −f(−x)

x
− 2n

f(−x)
x

,

Lemma 2.2. (i): The map

Mn(f)(x) = x
2nf(x)

is a topological isomorphism
• from E(R) onto En(R);
• from D(R) onto Dn(R).

(ii): For all f ∈ E(R),
Λα,n ◦Mn(f) = Mn ◦ Λα+2n(f),

where Λα+2n is the Dunkl operator of order α + 2n given by (1)
(iii): Let f ∈ En(R) and g ∈ Dn(R). Then

(7)
∫
R

Λα,nf(x)g(x)|x|2α+1dx = −
∫
R
f(x)Λα,ng(x)|x|2α+1dx.

2.3. Generalized Dunkl Transform. For λ ∈ C and x ∈ R put

(8) Ψλ,α,n(x) = x2neα+2n(iλx),

where eα+2n is the Dunkl kernel of index α + 2n given by (2).

Proposition 2.3. (i): Ψλ,α,n satisfies the differential-difference equation

(9) Λα,nΨλ,α,n = iλΨλ,α,n.

Definition 2.4. The generalized Dunkl transform of a function f ∈ Dn(R) is defined by

(10) Fα,n(f)(λ) =
∫
R
f(x)Ψ−λ,α,n(x)|x|2α+1dx, λ ∈ C.

Proposition 2.5. For every f ∈ Dn(R),
(11) Fα,n(Λα,nf)(λ) = iλFα,n(f)(λ),
Theorem 2.6. (i): For all f ∈ Dn(R), we have the inversion formula

f(x) = mα+2n

∫
R
Fα,n(f)(λ)Ψλ,α,n(x)|λ|2α+4n+1dλ,

where mα+2n is given by (5).
(ii): For every f ∈ Dn(R), we have the Plancherel formula

(12)
∫
R
|f(x)|2|x|2α+1dx = mα+2n

∫
R
|Fα,n(f)(λ)|2|λ|2α+4n+1dλ.



90 OZTURK

3. A Real Paley-Wiener Theorem

In this section, we give a short and simple proof of a real Paley-Wiener theorem for the
Dunkl transform.
We define the real Paley-Wiener space PWR(R) as the space of all f ∈ S(R) such that, for
N ∈ N0 = N∪{0}

(13) sup
x∈R,m∈N0

R−mm−N (1 + |x|)N|Λmα,nf(x)| < ∞.

Our real Paley-Wiener Theorem is the following:

Theorem 3.1. Let R > 0. The Generalized Dunkl transform Fα,n is a bijection from
PWR(R) onto C∞R (R), and by symmetry a bijection from C

∞
R (R) onto PWR(R).

Proof. Let f ∈ PWR(R), and λ outside [−R,R]. Then (7) and (9) yield

Fα,nf(λ) =
∫
R
f(x)Ψ−λ,α,n(x)|x|2α+1dx,

= (−iλ)−m
∫
R
f(x)Λmα,nΨ−λ,α,n(x)|x|

2α+1dx,

= (−iλ)−m(−1)m
∫
R

Λmα,nf(x)Ψ−λ,α,n(x)|x|
2α+1dx,

hence, for a positive C,

|Fα,nf(λ)| = |(−iλ)−m(−1)m
∫
R

Λmα,nf(x)Ψ−λ,α,n(x)|x|
2α+1dx|,

≤ |λ|−m
∫
R
|Λmα,nf(x)Ψ−λ,α,n(x)||x|

2α+1dx|,

≤ C|λ|−m
∫
R
RmmN (1 + |x|)−N|x|2α+2n+1dx|,

= C(
R

|λ|
)mmN

∫
R

(1 + |x|)−N|x|2α+2n+1dx|→ 0 for m →∞.

and thus SuppFα,nf ⊂ [−R,R].
Conversely, let f ∈C∞R (R). Fix N ∈ N0.

F−1α,nf(λ) := mα+2n
∫
R
f(λ)Ψλ,α,n(x)|λ|2α+4n+1dλ,

xN Λmα,nF
−1
α,nf(λ) = mα+2n

∫
R
f(λ)xN Λmα,nΨλ,α,n(x)|λ|

2α+4n+1dλ,

= mα+2n(−i)m
∫
R
λmf(λ)xN

x2n

λ2n
Ψx,α,n(λ)|λ|2α+4n+1dλ,

= (−i)mmα+2n
∫
R
λmf(λ)xN+2nΨx,α,n(λ)|λ|2α+2n+1dλ,

= (−i)m−N−2nmα+2n
∫
R
λmf(λ)ΛN+2nα,n Ψx,α,n(λ)|λ|

2α+2n+1dλ,

= (−i)m+N+2nmα+2n
∫
R

ΛN+2nα,n (λ
mf(λ))Ψx,α,n(λ)|λ|2α+2n+1dλ.

a small calculation give

Λα,n(λ
mf(λ) = mλm−1[f(λ) +

1

m
λ
d

dλ
f(λ) +

1

m
(α +

1

2
(f(λ) − (−1)mf(−λ)) −

2n

m

f(−λ)
λm

]



A REAL PALEY-WIENER THEOREM 91

Let f̃ denote the function in square bracket. An induction argument with f1 = f̃ and
f̃i+1 = f̃i, show that we can write, for m > N + 2n

Λα,n(λ
N+2n(λmf(λ)) = λm−N−2nmN+2nf̃N+2n(λ),

where f̃N+2n ∈C∞R (R) with suppf̃N+2n ⊂ suppf, and

‖ f̃N+2n ‖∞≤ C
N+2n∑
k=0

‖
dk

dxk
f ‖∞

where C is a positive constant only depending on f,α,n and N not on m. We get thus

|xN Λmα,nF
−1
α,nf(x)| ≤ Cmα+2nR

2α+2n+1mN
N+2n∑
k=0

‖
dk

dxk
f ‖∞

for all x ∈ R, and m > N + 2n, and thus F−1α,nf ∈ PWR(R)

4. A characterization of the support of the generalized Dunkl transform
on R

Theorem 4.1. Let 1 ≤ p ≤∞. Let f ∈S(R). Then

lim
m→∞

‖Λmα,nf‖
1
m
p,α,n = sup{|λ|,λ ∈ SuppFα,n(f)}.

Proof. Define Rf = sup{|λ|,λ ∈ SuppFα,n(f)}.
Assume that Fα,n has a compact support. Then f ∈ PWR(R) by Theorem 3.1 and

lim
m→∞

‖Λmα,nf‖
1
m
p,α,n ≤ Rf lim

m→∞
m

N
m = Rf

for all 1 ≤ p ≤∞, using (13) with N ≥ 2α + 2m + 3.
Now consider an arbitrary f ∈ .., using (7)

‖ Λmα,nf ‖
2
2,α,n =

∫
R
|Λmα,nf(x)|

2|x|2α+1dx,

=

∫
R

Λmα,nf(x)Λ
m
α,nf(x)|x|

2α+1dx,

= (−1)m
∫
R

Λ2mα,nf(x)f(x)|x|
2α+1dx.

Hölder’s inequality with 1
p

+ 1
q

= 1

(14) ‖Λmα,nf‖
2
2,α,n ≤‖Λ

2m
α,nf‖p,α,n‖f‖q,α,n.

Similarly, we get
‖Λm+1α,n f‖

2
2,α,n ≤‖Λ

2m+1
α,n f‖p,α,n‖Λα,nf‖q,α,n.

Let R < Rf. From (11) and (12)

‖ Λmα,nf ‖
2
2,α,n =

∫
R
|Λmα,nf(x)|

2|x|2α+1dλ,

= mα+2n

∫
R
|Fα,n(Λmα,nf(λ))|

2|λ|2α+4n+1dλ,

= mα+2n

∫
R
|λ|2m|Fα,nf(λ)|2|λ|2α+4n+1dλ,

≥ mα+2nR2m
∫
R
|Fα,nf(λ)|2|λ|2α+4n+1dλ,



92 OZTURK

where the last integral is positive.
Combining (14) with the above inequality yields

lim inf
m→∞

‖Λ2mα,nf‖
1

2m
p,α,n ≥ lim inf

m→∞
‖Λmα,nf‖

1
m
2,α,n ≥ R

for any 1 ≤ p ≤∞, and similarly

lim inf
m→∞

‖Λ2m+1α,n f‖
1

2m+1
p,α,n ≥ Rf.

We thus conclude, for any 0 < R < Rf

R ≤ lim inf
m→∞

‖Λmα,nf‖
1
m
p,α,n ≤ lim sup

m→∞
‖Λmα,nf‖

1
m
p,α,n ≤ Rf

this complect the proof of the theorem.

References

[1] Al Sadhan, S.A., Al Subaie, R.F. and Mourou, M.A. Harmonic Analysis Associated with A First-Order
Singular Differential-Difference Operator on the Real Line. Current Advances in Mathematics Research,
1(2014), 23-34.

[2] M.A. Mourou and K. Trimèche, "Transmutation operators and Paley-Wiener theorem associated with
a singular Differential-Difference operator on the real line", Analysis and Applications, 1(2003), 43-70.

[3] Nils Byrial Andersen Real Paley-Wiener theorems for the Dunkl transform on R, Integral Transforms
and Special Functions, 17(2006), 543-547

[4] Chettaoui, C., Trimèche, K., New type Paley-Wiener theorems for the Dunkl transform on R. Integral
Transforms and Special Functions, 14(2003), 97-115.

[5] Trimèche, K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators. Integral
Transforms and Special Functions, 13(2002), 17-38.

Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II,
Casablanca, Morocco

∗Corresponding author