International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 1 (2014), 53-62
http://www.etamaths.com

ON ESTIMATES FOR THE DUNKL TRANSFORM IN THE

SPACE L2,α(R)

MOHAMED EL HAMMA∗ AND RADOUAN DAHER

Abstract. In this paper, we study two estimates useful in applications are

proved for the Dunkl transform in a Hilbert space L2,α(R) = L2(R, |x|2α+1dx), α >
−1

2
as applied to some classes of functions characterized by a generalized mod-

ulus of continuity.

1. Introduction and preliminaries

Dunkl operators are differential-difference operators introduced in 1989, by Dunkl
[5]. On the real line, these operators, which are denote by Dα, depend on a real
parameter α > −1

2
and they are associated with the reflection group Z2 on R .

For α > −1
2
, Dunkl kernel eα is defined as the unique solution of a differential-

difference equation related to Dα and satisfying eα(0) = 1. This kernel is used
to define Dunkl transform which was introduced by Dunkl in [6]. More complete
results concerning this transform were later obtained by de Jeu [7]. Rösler in [8]
shows that Dunkl kernels verify a product formula. This allows us to define Dunkl
translation operators Th, h ∈ R.

The Dunkl operator on R of index (α + 1
2
) is defined in [5] by

Df(x) = Dαf(x) =
df(x)

dx
+ (α +

1

2
)
f(x) −f(−x)

x
, α >

−1
2
.

These operators are very important in mathematics and physics.

In this paper, we prove two useful estimates in certain classes of functions char-
acterized by a generalized continuity modulus and connected with the Dunkl trans-
form in L2,α(R), For this purpose, we use a translation operator in [4]. We point
out that similar results have been established in the context of Fourier transform
in real line (see [2]).

Assume that L2,α(R), is stand for the Hilbert space which consists of measurable
functions f(x) is defined on R with the finite norm

‖f‖ = ‖f‖2,α =
(∫ +∞
−∞

|f(x)|2|x|2α+1dx
)1

2

,

2010 Mathematics Subject Classification. 44A15; 33C52.
Key words and phrases. Dunkl operator; generalized modulus of continuity; Dunkl transform.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

53



54 HAMMA AND DAHER

Given a function f ∈ L2,α(R), the Dunkl transform [4] of order α is defined as

f̂(λ) =

∫ +∞
−∞

f(x)eα(λx)|x|2α+1dx, λ ∈ R,

where eα(x) Dunkl kernel is defined by

(1) eα(x) = jα(x) + i(2α + 2)
−1xjα+1(x).

The function y = eα(x) satisfies the equation Dy = iy with the initial condition
y(0) = 1, jα(x) is a normalized Bessel function of the first kind, i.e

(2) jα(x) =
2αΓ(α + 1)Jα(x)

xα
,

where Jα(x) is a Bessel function of the first kind ([3], chap7) the function jα is
infinitely differentiable and even, in addition, jα(0) = 1.

From formula (1), we have

(3) |1 − jα(x)| ≤ |1 −eα(x)|

The inverse Dunkl transform is defined by the formula

f(x) =
1

(2α+1Γ(α + 1))2

∫ +∞
−∞

f̂(λ)eα(−λx)|λ|2α+1dλ.

In L2,α(R), we define the operator of the generalized Dunkl translation (see [9])

Thf(x) = C(

∫ π
0

fe(G(x,h,ϕ))h
e( x ,h,ϕ)sin2αϕdϕ

+

∫ π
0

f0(G(x,h,ϕ))h
0(x,h,ϕ)sin2αϕdϕ)

where

C =
Γ(α + 1)

Γ( 1
2
)Γ(α + 1

2
)
, G(x,h,ϕ) =

√
x2 + h2 − 2|xh|cosϕ

he(x,h,ϕ) = 1 −sgn(xh)cosϕ
and {

h0(x,h,ϕ) =
(x+h)he(x,h,ϕ)

G(x,h,ϕ)
for (x,h) 6= (0, 0)

h0(x,h,ϕ) = 0 for (x,h) = (0, 0)

fe(x) =
1

2
(f(x) + f(−x)), f0(x) =

1

2
(f(x) −f(−x)).

Lemma 1.1. [4] Let f ∈ L2,α(R), then the following equality is true for any h ∈ R

(̂Thf)(λ) = eα(λh)f̂(λ).



DUNKL TRANSFORM IN THE SPACE L2,α(R) 55

The first-and higher order finite differences of f(x) are defined as follows

∆hf(x) = Thf(x) −f(x) = (Th − I)f(x),
where I is the identity operator in L2,α(R).

∆khf(x) = ∆h(∆
k−1
h f(x)) = (Th − I)

kf(x) =

k∑
i=0

(−1)k−i(ki )T
i
hf(x),

where T0hf(x) = f(x), T
i
hf(x) = Th(T

i−1
h f(x)) for i=1,2,....,k and k=1,2,.....

The kth order generalized modulus of continuity of function f ∈ L2,α(R) such
that

Ωk(f,δ) = sup
0<h≤δ

‖∆khf(x)‖.

Let Wm2,α be the Sobolev space constructed by the operator D such that

Wm2,α = {f ∈ L2,α(R), D
jf ∈ L2,α(R), j = 1, 2, .....,m}

W
m,k
2,φ (D) denote the class of functions f ∈ W

m
2,α, satistying the estimate

Ωk(D
mf,δ) = O(φ(δm)),

where φ(t) is any nonnegative function given on [0,∞) and φ(0) = 0, for the Dunkl
operator D, we have D0f = f, Dmf = D(Dm−1f),m = 1, 2, ...

From lemma 1.1, we have

Thf(x) =
1

(2α+1Γ(α + 1))2

∫ +∞
−∞

eα(λh)f̂(λ)eα(−λx)|λ|2α+1dλ

Therefore, combining the relation

Thf(x) −f(x) =
1

(2α+1Γ(α + 1))2

∫ +∞
−∞

(eα(λh) − 1)f̂(λ)eα(−λx)|λ|2α+1dλ

Parseval’s identity gives (see [4])

‖Thf(x) −f(x)‖2 = A
∫ +∞
−∞

|eα(λh) − 1|2|f̂(λ)|2|λ|2α+1dλ,

where

A = (2α+1Γ(α + 1))−2.

Hence, for any function f ∈ Wm,k2,φ (D), we obtain

(4) ‖∆khD
mf(x)‖2 = A

∫ +∞
−∞

|eα(λh) − 1|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ.



56 HAMMA AND DAHER

2. Estimates for the Dunkl transform

Taking into account what was said in section 1, for some classes of functions
characterized by the generalized modulus of continuity, we can prove two estimates
for the integral

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ,

which are useful in applications.

Lemma 2.1. For x ∈ R the following inequalities are fulfilled:

(1) |eα(x)| ≤ 1,
(2) |1 −eα(x)| ≤ 2|x|.

Proof. (see[4])

Lemma 2.2. The following inequalities are fulfilled:

(1) 1 − jp(x) = O(1), x ≥ 1,
(2) 1 − jp(x) = O(x2), 0 ≤ x ≤ 1,
(3)
√
hxJp(hx) = O(1), hx ≥ 0.

Proof. (see [1])

Theorem 2.3. For f(x) ∈ L2,α(R)

sup
W
m,k
2,φ

(D)

√∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ = O(R−mφ(
c

R
)k),

where m=0,1,....; k=1,2....; c > 0 is a fixed constant, and φ(t) is any nonnegative
function defined on [0,∞).

Proof. Let f ∈ Wm,k2,φ (D). Taking into account the Hölder inequality yields



DUNKL TRANSFORM IN THE SPACE L2,α(R) 57

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ−
∫
|λ|≥R

jα(λh)|f̂(λ)|2|λ|2α+1dλ

=

∫
|λ|≥R

(1 − jα(λh))|f̂(λ)|2|λ|2α+1dλ

=

∫
|λ|≥R

(1 − jα(λh))(|f̂(λ)||λ|α+
1
2 )2dλ

=

∫
|λ|≥R

(1 − jα(λh))(|f̂(λ)||λ|α+
1
2 )2−

1
k (|f̂(λ)||λ|α+

1
2 )

1
k dλ

≤ (
∫

|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k (

∫
|λ|≥R

|1 − jα(λh)|2k|f̂(λ)|2|λ|2α+1dλ)
1
2k

≤ (
∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k (

∫
|λ|≥R

|λ|−2m|1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ)
1
2k

≤ R
−m
k (

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k (

∫
|λ|≥R

|1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ)
1
2k

In view of (4), we have

∫
|λ|≥R

|λ|2m|1 −eα(λh)|2k|f̂(λ)|2|λ|2α+1dλ ≤
1

A
‖∆khD

mf(x)‖2.

Therefore

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ ≤
∫

|λ|≥R

jα(λh)|f̂(λ)|2|λ|2α+1dλ

+
1

A
R

−m
k (

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k .‖∆khD

mf(x)‖
1
k

In view of formulas (2) and (2) in lemma 2.2,

jα(λh) = O((|λh|)−α−
1
2 ).

consequently



58 HAMMA AND DAHER

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ = O(
∫

|λ|≥R

|hλ|−α−
1
2 |f̂(λ)|2|λ|2α+1dλ)

+R
−m
k (

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k ‖∆khD

mf(x)‖
1
k

= O((Rh)−α−
1
2 )

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ

+R
−m
k (

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k ‖∆khD

mf(x)‖
1
k

or

(1 −O(Rh)−α−
1
2 )

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ

= O(R
−m
k )(

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k ‖∆khD

mf(x)‖
1
k

Setting h = c
R

in the last inequality and choosing c > 0 such that

1 −O(c−α−
1
2 ) ≥

1

2
We obtain

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ = O(R
−m
k )(

∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ)
2k−1
2k φ

1
k ((

c

R
)k)

we have ∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ = O(R−2mφ2((
c

R
)k)).

the theorem is proved.

Theorem 2.4. Let φ(t) = tν, then
 ∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ




1
2

= O(R−m−kν) ⇐⇒ f ∈ Wm,k2,φ (D)

where m=0,1,...; k=1,2,....; 0 < ν < 2.

Proof. Sufficiency by Theorem 2.3 let f ∈ Wm,k2,tν (D) we have


 ∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ




1
2

= O(R−m−kν)



DUNKL TRANSFORM IN THE SPACE L2,α(R) 59

Necessity: Let

√√√√ ∫
|λ|≥R

|f̂(λ)|2|λ|2α+1dλ = O(R−m−kν)

that is

∫
|λ|≥R

|f̂(ξ)|2|λ|2α+1dλ = O(R−2m−2kν)

It is easy to prove, that there exists a function f ∈ L2,α(R) such that Dmf ∈
L2,α(R) and

Dmf(x) =
(−i)m

(2α+1Γ(α + 1))2

∫ +∞
−∞

λmf̂(λ)eα(−λx)|λ|2α+1dλ

Then, we have the equality

‖∆khD
mf(x)‖2 = A

∫ +∞
−∞

|1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ

This integral is divided into two:

∫ +∞
−∞

=

∫
|λ|<R

+

∫
|λ|≥R

= I1 + I2

where R = [h−1] . and estimate each of them.

From (1) in lemma 2.1, we have



60 HAMMA AND DAHER

I2 =

∫
|λ|≥R

|1 −eα(λh)|2k|f̂(λ)|2|λ|2α+2m+1dλ

= O


 ∫
|λ|≥R

|f̂(λ)|2|λ|2α+2m+1dλ




= O

(
∞∑
n=R

∫ n+1
n

|f̂(λ)|2|λ|2α+2m+1dλ

)

= O

(
∞∑
n=R

n2m
∫ n+1
n

|f̂(λ)|2|λ|2α+1dλ

)

= O

(
∞∑
n=R

n2m
[∫ ∞

n

|f̂(λ)|2|λ|2α+1dλ−
∫ ∞
n+1

|f̂(λ)|2|λ|2α+1dλ
])

= O

(
∞∑
n=R

n2m
∫ ∞
n

|f̂(λ)|2|λ|2α+1dλ−
∞∑
n=R

n2m
∫ ∞
n+1

|f̂(λ)|2|λ|2α+1dλ

)

= O

(
R2m

∫ ∞
R

|f̂(λ)|2|λ|2α+1dλ

+

∞∑
n=R+1

n2m
∫ ∞
n

|f̂(λ)|2|λ|2α+1dλ−
∞∑
n=R

n2m
∫ ∞
n+1

|f̂(λ)|2|λ|2α+1dλ

)

= O(R2m
∫ ∞
R

|f̂(λ)|2|λ|2α+1dλ +
∞∑
n=R

((n + 1)2m −n2m)
∫ ∞
n

|f̂(λ)|2|λ|2α+1dλ)

= O(R2m
∫ ∞
R

|f̂(λ)|2|λ|2α+1dλ +
∞∑
n=R

n2m−1
∫ ∞
n

|f̂(λ)|2|λ|2α+1dλ)

= O(R2mR−2m−2kν) + O(

∞∑
n=R

n2m−1n−2m−2kν)

= O(R−2kν) + O(R−2kν) = O(h2kν)

i.e

I2 = O(h
2kν).

We estimate I1, since (2) in lemma 2.1, we have



DUNKL TRANSFORM IN THE SPACE L2,α(R) 61

I1 =

∫
|λ|<R

|f̂(λ)|2|λ|2α+2m+1|1 −eα(λh)|2kdλ

= O(h2k)

∫
|λ|<R

|f̂(λ)|2|λ|2α+2m+2k+1dλ

= O(h2k)

R∑
n=0

∫
n≤|λ|<n+1

|f̂(λ)|2|λ|2α+2m+2k+1dλ

= O(h2k)

R∑
n=0

(n + 1)2m+2k
∫

n≤|λ|<n+1

|f̂(λ)|2|λ|2α+1dλ

= O(h2k)

R∑
n=0

(n + 1)2m+2k


 ∫
|λ|≥n

|f̂(λ)|2|λ|2α+1dλ−
∫

|λ|≥n+1

|f̂(λ)|2|λ|2α+1dλ




= O(h2k)


1 + R∑

n=0

((n + 1)2m+2k −n2m+2k)
∫
|λ|≥n

|f̂(λ)|2|λ|2α+1dλ




= O(h2k)


1 + R∑

n=0

n2m+2k−1
∫
|λ|≥n

|f̂(λ)|2|λ|2α+1dλ




= O(h2k)

[
1 +

R∑
n=0

n2m+2k−1n−2m−2kν

]

= O(h2k)

[
1 +

R∑
n=0

n2k−2kν−1

]
= O(h2k)O(R2k−2kν) = O(h2kν)

that is

I1 = O(h
2kν)

Combining the estimates for I1 and I2 gives

‖∆khD
mf(x)‖ = O(hkν)

The necessity is proved

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62 HAMMA AND DAHER

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

∗Corresponding author