International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 1 (2015), 15-21
http://www.etamaths.com

AN ANALOG OF TITCHMARSH’S THEOREM FOR THE

JACOBI-DUNKL TRANSFORM IN THE SPACE L2α,β(R)

A. ABOUELAZ, A. BELKHADIR∗ AND R. DAHER

Abstract. In this paper, using a generalized Jacobi-Dunkl translation oper-

ator, we prove an analog of Titchmarsh’s theorem for functions satisfying the

Jacobi-Dunkl Lipschitz condition in L2(R,Aα,β(t)dt) , α ≥ β ≥−12 ,α 6= −
1
2
.

1. Introduction

Titchmarsh’s theorem characterizes the set of functions satisfying the Cauchy-
Lipschitz condition by means of an asymptotic estimate growth of the norm of their
Fourier transform, namely we have:

Theorem 1.1. [10] Let α ∈ (0, 1) and assume that f ∈ L2(R) . Then the
following are equivalents:

(1) ‖f(t + h) −f(t)‖ = O(hα) , as α → 0 ;

(2)

∫
|λ|≥r

|f̂(λ)|2dλ = O(r−2α) , as r →∞ .

where f̂ stands for the Fourier transform of f .

In this paper, we prove an analog of Theorem 1.1 for the Jacobi-Dunkl trans-
form for functions satisfying the Jacobi-Dunkl Lipschitz condition in the space
L2(R,Aα,β(t)dt) . For this purpose, we use the generalized translation operator.
Similar results have been established in the context of noncompact rank one Rie-
mannian symetric spaces [9].

In section 2 below, we recapitulate from [1, 2, 3, 5] some results related to the
harmonic analysis associated with Jacobi-Dunkl operator Λα,β .

Section 3 is devoted to the main result after defining the class Lip(δ, 2,α,β)
of functions in L2α,β(R) satisfying the Lipschitz condition correspondent to the
generalized Jacobi-Dunkl translation.

2. Notations and preliminaries

The Jacobi-Dunkl function with parameters (α,β) , α ≥ β ≥ −1
2
,α 6= −1

2
, is

defined by the formula :

(1) ∀x ∈ R, ψ(α,β)λ (x) =

{
ϕ
(α,β)
µ (x) −

i

λ

d

dx
ϕ
(α,β)
µ (x) , if λ ∈ C\{0};

1 , if λ = 0.

2010 Mathematics Subject Classification. 33C45.
Key words and phrases. Titchmarsh’s theorem; Jacobi-Dunkl transform; generalized Jacobi-

Dunkl translation; Jacobi-Dunkl Lipschitz condition.

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.

15



16 ABOUELAZ, BELKHADIR AND DAHER

with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕ
(α,β)
µ is the Jacobi function given by:

(2) ϕ(α,β)µ (x) = F

(
ρ + iµ

2
,
ρ− iµ

2
; α + 1,−(sinh(x))2

)
,

F is the Gauss hypergeometric function (see [1, 6, 7]).

ψ
(α,β)
λ is the unique C

∞-solution on R of the differentiel-difference equation

(3)

{
Λα,βU = iλU , λ ∈ C;
U(0) = 1.

where Λα,β is the Jacobi-Dunkl operator given by:

Λα,βU(x) =
dU
dx

(x) + [(2α + 1) coth x + (2β + 1) tanh x] ×
U(x) −U(−x)

2

The operator Λα,β is a particular case of the operator D given by:

DU(x) =
dU
dx

(x) +
A′(x)

A(x)

(
U(x) −U(−x)

2

)
where A(x) = |x|2α+1B(x) , and B a function of class C∞ on R , even and positive.
The operator Λα,β corresponds to the function

A(x) = Aα,β(x) = 2
ρ(sinh |x|)2α+1(cosh |x|)2β+1.

Using the relation

d

dx
ϕ(α,β)µ (x) = −

µ2 + ρ2

4(α + 1)
sinh(2x)ϕ(α+1,β+1)µ (x) ,

the function ψ
(α,β)
λ can be written in the form above (See [2]),

(4) ψ
(α,β)
λ (x) = ϕ

(α,β)
µ (x) + i

λ

4(α + 1)
sinh(2x)ϕ(α+1,β+1)µ (x) , ∀x ∈ R ,

where λ2 = µ2 + ρ2 , ρ = α + β + 1.

Denote by L2α,β(R) = L
2(R,Aα,β(t)dt) the space of measurable functions g on R

such that

||g||L2
α,β

(R) =

(∫
R
|g(t)|2Aα,β(t)dt

)1/2
< +∞

.
Using the eigenfunctions ψ

(α,β)
λ of the operator Λα,β called the Jacobi-Dunkl

kernels, we define the Jacobi-Dunkl transform of a function f ∈ L2α,β(R) by:

(5) Fα,β(f)(λ) =
∫
R
f(x)ψ

(α,β)
λ (x)Aα,β(x)dx, ∀λ ∈ R .

and the inversion formula

(6) f(t) =

∫
R
Fα,β(f)(λ)ψ

(α,β)
−λ (t)dσ(λ) ,

where: dσ(λ) =
|λ|

8π
√
λ2 −ρ2|Cα,β(

√
λ2 −ρ2)|

IR\]−ρ,ρ[(λ)dλ



AN ANALOG OF TITCHMARSH’S THEOREM FOR JACOBI-DUNKL TRANSFORM 17

Here, Cα,β(µ) =
2ρ−iµΓ(α + 1)Γ(iµ)

Γ( 1
2
(ρ + iµ))Γ( 1

2
(α−β + 1 + iµ))

, µ ∈ C\ (iN) .

and IR\]−ρ,ρ[ is the characteristic function of R\] −ρ,ρ[ .
Denote L2σ(R) = L

2(R,dσ(λ)).
The Jacobi-Dunkl transform is a unitary isomorphism from L2α,β(R) onto L

2
σ(R),

i.e.

(7) ||f|| = ||f||L2
α,β

(R) = ||Fα,β(f)||L2σ(R) .

The operator of Jacobi-Dunkl translation is defined by:

(8) Txf(y) =

∫
R
f(z)dνα,βx,y (z) , ∀ x,y ∈ R .

where να,βx,y , x,y ∈ R are the signed measures given by

(9) dνα,βx,y (z) =




Kα,β(x,y,z)Aα,β(z)dz , if x,y ∈ R?;
δx , if y = 0;
δy , if x = 0.

Here, δx is the Dirac measure at x.
And,

Kα,β(x,y,z) = Mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αIIx,y ×
∫π
0
ρθ(x,y,z)

×(gθ(x,y,z))
α−β−1
+ sin

2β θdθ.

Ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x| + |y||, |x| + |y|] ,

ρθ(x,y,z) = 1 −σθx,y,z + σ
θ
z,x,y + σ

θ
z,y,x

σθx,y,z =




cosh(x) + cosh(y) − cosh(z) cos(θ)
sinh(x) sinh(y)

, if xy 6= 0;

0 , if xy = 0.
, ∀x,y,z ∈ R , ∀θ ∈ [0,π] .

gθ(x,y,z) = 1 − cosh2 x− cosh2 y − cosh2 z + 2 cosh x cosh y cosh z cos θ .

t+ =

{
t , if t > 0;
0 , if t ≤ 0.

and,

Mα,β =




2−2ρΓ(α + 1)
√
πΓ(α−β)Γ(β + 1

2
)

, if α > β;

0 , if α = β.

In [2], we have

(10) Fα,β(Thf)(λ) = ψ
α,β
λ (h).Fα,β(f)(λ) ; h,λ ∈ R .

For α ≥ −1
2

, we introduce the bessel normalized function of the first kind defined
by

jα(z) = Γ(α + 1)

∞∑
n=0

(−1)n(z
2
)2n

n!Γ(n + α + 1)
, z ∈ C.



18 ABOUELAZ, BELKHADIR AND DAHER

Moreover, we see that lim
z→0

jα(z) − 1
z2

6= 0 , by consequence, there exists C1 > 0
and η > 0 satisfying

(11) |z| ≤ η ⇒ |jα(z) − 1| ≥ C1|z|2 .

Lemma 2.1. The following inequalities are valids for Jacobi functions ϕα,βµ (t) :

(1) |ϕ(α,β)µ (t)| ≤ 1 ;
(2) |1 −ϕ(α,β)µ (t)| ≤ t2(µ2 + ρ2) .

Proof. (See [8], Lemma 3.1-3.2) �

Lemma 2.2. Let α ≥ β ≥ −1
2

, α 6= −1
2

. Then for |ν| ≤ ρ , there exists a
positive constant C2 such that

|1 −ϕ(α,β)µ+iν (t)| ≥ C2|1 − jα(µt)| .

Proof. (See [4], Lemma 9) �

3. Main result

In this section we introduce and prove an analog of theorem 1.1. Firstly we have
to define, for functions in L2α,β(R) , the condition of Cauchy-Lipschitz related to
the Jacobi-Dunkl translation operator given in (8).

Definition 3.1. Let δ ∈ (0, 1) . A function f ∈ L2α,β(R) is said to be in the
Jacobi-Dunkl-Lipschitz class, denoted by Lip(δ, 2,α,β) , if ||Thf + T−hf − 2f|| =
O(hδ) , as h → 0 .

Theorem 3.2. Let f ∈ L2α,β(R) . Then the following are equivalents:
(1) f ∈ Lip(δ, 2,α,β) ;

(2)

∫
|λ|≥r

|Fα,β(f)(λ)|2dσ(λ) = O(r−2δ) , as r →∞ .

Proof. 1) ⇒ 2) . Assume that f ∈ Lip(δ, 2,α,β); then we have:

||Thf + T−hf − 2f|| = O(hδ) , as h → 0 .

Fα,β(Thf + T−hf − 2f)(λ) = (ψ
(α,β)
λ (h) + ψ

(α,β)
λ (−h) − 2).Fα,β(f)(λ).

Since ψ
(α,β)
λ (h) = ϕ

(α,β)
µ (h) + i

λ

4(α + 1)
sinh(2h)ϕ

(α+1,β+1)
µ (h),

ψ
(α,β)
λ (−h) = ϕ

(α,β)
µ (−h) − i

λ

4(α + 1)
sinh(2h)ϕ

(α+1,β+1)
µ (−h),

and ϕ
(α,β)
µ is even [See (2)]; then:

Fα,β(Thf + T−hf − 2f)(λ) = 2(ϕ(α,β)µ (h) − 1).Fα,β(f)(λ).
From Parseval’s identity (7) we write:

(12) ||Thf + T−hf − 2f||2 = 4
∫
R
|1 −ϕ(α,β)µ (h)|

2|Fα,β(f)(λ)|2dσ(λ).

By (11) and lemma 2.2, we get:



AN ANALOG OF TITCHMARSH’S THEOREM FOR JACOBI-DUNKL TRANSFORM 19

∫
η
2h
≤|λ|≤η

h

|1−ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) ≥

∫
η
2h
≤|λ|≤η

h

C21C
2
2|µh|

4|Fα,β(f)(λ)|2dσ(λ) ,

From
η

2h
≤ |λ| ≤

η

h
we have,( η

2h

)2
−ρ2 ≤ µ2 ≤

(η
h

)2
−ρ2

⇒ µ2h2 ≥
η2

4
−ρ2h2

Take h ≤
η

3ρ
, then we have µ2h2 ≥ C3 = C3(η).

So,

∫
η
2h
≤|λ|≤η

h

|1−ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) ≥ C21C

2
2C

2
3

∫
η
2h
≤|λ|≤η

h

|Fα,β(f)(λ)|2dσ(λ) .

There exists then a positive constant C such that:∫
η
2h
≤|λ|≤η

h

|Fα,β(f)(λ)|2dσ(λ) ≤ C
∫
R
|1 −ϕ(α,β)µ (h)|

2|Fα,β(f)(λ)|2dσ(λ)

≤ Ch2δ ,
For all 0 < h ≤

η

3ρ
, (see (12)). Then we have,∫

r≤|λ|≤2r
|Fα,β(f)(λ)|2dσ(λ) ≤ Cr−2δ , as r →∞.

Furthermore, we obtain:

∫
|λ|≥r

|Fα,β(f)(λ)|2dσ(λ) =
∞∑
i=0

∫
2ir≤|λ|≤2i+1r

|Fα,β(f)(λ)|2dσ(λ)

≤ C
∞∑
i=0

(
2ir
)−2δ

≤ Cr−2δ.
This proves that:∫

|λ|≥r
|Fα,β(f)(λ)|2dσ(λ) = O(r−2δ) , as r →∞.

2) ⇒ 1) . Suppose now that∫
|λ|≥r

|Fα,β(f)(λ)|2dσ(λ) = O(r−2δ) , as r →∞,

and write

∫
R
|1 −ϕ(α,β)µ (h)|

2|Fα,β(f)(λ)|2dσ(λ) =
∫
|λ|< 1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ)

+

∫
|λ|≥1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ)



20 ABOUELAZ, BELKHADIR AND DAHER

— Using the inequality (1) of lemma 2.1, we get:

∫
|λ|≥1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) ≤ 4

∫
|λ|≥1

h

|Fα,β(f)(λ)|2dσ(λ)

then,

(13)

∫
|λ|≥1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) = O(h2δ) , as h → 0.

— Set φ(x) =

∫ ∞
x

|Fα,β(f)(λ)|2dσ(λ) .
An integration by parts gives:∫ x

0

λ2|Fα,β(f)(λ)|2dσ(λ) =
∫ x
0

−λ2φ′(λ)dλ

= −x2φ(x) + 2
∫ x
0

λφ(λ)dλ

≤ 2
∫ x
0

O(λ1−2δ)dλ

= O(x2−2δ).

From the second inequality of lemma 2.1, we get

∫
|λ|< 1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) ≤

∫
|λ|< 1

h

(µ2 + ρ2)h2|Fα,β(f)(λ)|2dσ(λ)

≤ h2
∫
|λ|< 1

h

λ2|Fα,β(f)(λ)|2dσ(λ)

= O(h2.h−2+2δ).

Hence,

(14)

∫
|λ|< 1

h

|1 −ϕ(α,β)µ (h)|
2|Fα,β(f)(λ)|2dσ(λ) = O(h2δ).

Finally, we conclude from (13) and (14) that∫
R
|1 −ϕ(α,β)µ (h)|

2|Fα,β(f)(λ)|2dσ(λ) =
∫
|λ|< 1

h

+

∫
|λ|≥1

h

= O(h2δ) + O(h2δ)

= O(h2δ).

And this ends the proof. �

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Department of Mathematics and Informatics, Faculty of Science Ain Chock, Univer-

sity of Hassan II, Casablanca, Morocco,

∗Corresponding author