Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611 CJMS
.
5(2)(2016), 98-105
Some Results for the Jacobi-Dunkl Transform in the
Space
L
p(
R
, A
α,β
(
x
)
dx
)
S. EL OUADIH
1and R. DAHER
21
Department of Mathematics, Faculty of Sciences A¨ın Chock,
University Hassan II, Casablanca, Morocco
2
Department of Mathematics, Faculty of Sciences A¨ın Chock,
University Hassan II, Casablanca, Morocco
Abstract. In this paper, using a generalized Jacobi-Dunkl trans-lation operator, we obtain a generalization of Titchmarsh’s theorem for the Dunkl transform for functions satisfying the (ϕ, p)-Lipschitz Jacobi-Dunkl condition in the spaceLp(R, Aα,β(x)dx), α≥β≥−21,
α̸= −1 2 .
Keywords: Jacobi-Dunkl operator, Jacobi-Dunkl transform, gen-eralized Jacobi-Dunkl translation.
2000 Mathematics subject classification: 42B37.
1.
Introduction and Preliminaries
Titchmarsh’s ([8], Theorem 85) characterized the set of functions in
L
2(
R
) satisfying the Cauchy-Lipschitz Condition by means of an
asymp-totic estimate growth of the norm of their Fourier transform, namely we
have:
Theorem 1.1.
[8]
Let
α
∈
(0
,
1)
and assume that
f
∈
L
2(
R
)
. Then the
following are equivalents:
(a)
∥
f
(
x
+
h
)
−
f
(
x
)
∥
=
O
(
h
α)
,
as
h
→
0
,
1Corresponding author: [email protected] Received: 1 July 2015
Revised: 24 July 2016 Accepted: 6 August 2016
(b)
∫
|λ|≥r
|
f
b
(
λ
)
|
2dλ
=
O
(
r
−2α)
,
as
r
→ ∞
,
where
f
b
stands for the Fourier transform of
f
.
A similar result of Theorem 1.1 has been established for the Jacobi
transform in the space
L
2(
R
, A
α,β(
x
)
dx
) (see [11]). In this paper, we
prove a generalization of Theorem 1.1 for the Jacobi-Dunkl transform
for functions satisfying the (
ϕ, p
)-Lipschitz Jacobi-Dunkl condition in
the space
L
p(
R
, A
α,β
(
x
)
dx
)
,
1
< p
≤
2. For this purpose, we use the
generalized Jacobi-Dunkl translation operator.
In this section, we recapitulate from ([1]-[6]) some results related to
the harmonic analysis associated with Jacobi-Dunkl operator Λ
α,β.
The Jacobi-Dunkl function with parameters (
α, β
)
, α
≥
β
≥
−21, α
̸
=
−21,
is defined by the formula:
∀
x
∈
R
, ψ
λα,β(
x
) =
φ
α,βµ(
x
)
−
λidxdφ
α,βµ(
x
)
,
if
λ
∈
C\{
0
}
,
1
,
if
λ
= 0,
with
λ
2=
µ
2+
ρ
2,
ρ
=
α
+
β
+ 1 and
φ
α,βµ
is the Jacobi function given
by:
φ
α,βµ(
x
) =
F
(
ρ
+
iµ
2
,
ρ
−
iµ
2
, α
+ 1
,
−
(sinh(
x
))
2
)
,
F is the Gauss hypergeometric function (see [1],[7]).
ψ
λα,βis the unique
C
∞-solution on
R
of the differential-difference
equa-tion
Λ
α,βU
=
iλ
U
,
λ
∈
C
,
U
(0) = 1
,
(see [5]), where Λ
α,βis the Jacobi-Dunkl operator given by:
Λ
α,βU
(
x
) =
d
U
(
x
)
dx
+
A
′α,β(
x
)
A
α,β(
x
)
×
(
U
(
x
)
− U
(
−
x
)
2
)
,
with
A
α,β(
x
) = 2
ρ(sinh
|
x
|
)
2α+1(cosh
|
x
|
)
2β+1,
i.e.,
Λ
α,βU
(
x
) =
d
U
(
x
)
dx
+ [(2
α
+ 1) coth
x
+ (2
β
+ 1) tanh
x
]
×
U
(
x
)
− U
(
−
x
)
2
.
Using the relation
d
dx
φ
α,β
µ
(
x
) =
−
µ
2+
ρ
24(
α
+ 1)
sinh(2
x
)
φ
the function
ψ
α,βλcan be written in the form above (see [2])
ψ
λα,β(
x
) =
φ
α,βµ(
x
) +
i
λ
4(
α
+ 1)
sinh
(2
x
)
φ
α+1,β+1
µ
(
x
)
,
x
∈
R
,
where
λ
2=
µ
2+
ρ
2,
ρ
=
α
+
β
+ 1.
Denote
L
pα,β(
R
) =
L
pα,β(
R
, A
α,β(
x
)
dx
)
,
1
< p
≤
2 the space of
measur-able functions
f
on
R
such that
∥
f
∥
p,α,β=
(∫
R
|
f
(
x
)
|
p
A
α,β
(
x
)
dx
)
1/p<
+
∞
.
Using the eigenfunctions
ψ
α,βλof the operator Λ
α,βcalled the
Jacobi-Dunkl kernels, we define the Jacobi-Jacobi-Dunkl transform by
F
α,βf
(
λ
) =
∫
R
f
(
x
)
ψ
α,β
λ
(
x
)
A
α,β(
x
)
dx,
λ
∈
R
,
and the inversion formula by
f
(
t
) =
∫
R
F
α,βf
(
λ
)
ψ
α,β
−λ
(
t
)
dσ
(
λ
)
,
where
dσ
(
λ
) =
|
λ
|
8
π
√
λ
2−
ρ
2|
C
α,β
(
√
λ
2−
ρ
2)
|
I
R\]−ρ,ρ[(
λ
)
dλ.
Here,
C
α,β(
µ
) =
2
ρ−iµΓ(
α
+ 1)Γ(
iµ
)
Γ(
12(
ρ
+
iµ
))Γ(
12(
α
−
β
+ 1 +
iµ
))
,
µ
∈
C\
(
i
N
)
,
and
I
R\]−ρ,ρ[is the characteristic function of
R\
]
−
ρ, ρ
[.
The Jacobi-Dunkl transform is a unitary isomorphism from
L
2α,β(
R
) onto
L
2(
R
, dσ
(
λ
)), i.e.,
∥
f
∥
2,α,β=
∥F
α,β(
f
)
∥
L2(R,dσ(λ)).
(1.1)
Plancherel’s theorem (1.1) and the Marcinkiewics interpolation theorem
(see [8]) we get for
f
∈
L
pα,β(
R
) with 1
< p
≤
2 and
q
such that
1p+
1q= 1,
∥F
α,β(
f
)
∥
Lq(R,dσ(λ))≤
K
∥
f
∥
p,α,β,
(1.2)
where
K
is a positive constant (see [6]).
The operator of Jacobi-Dunkl translation is defined by :
T
xf
(
y
) =
∫
R
f
(
z
)
dν
α,β
where
ν
x,yα,β(
z
)
,
x, y
∈
R
are the signed measures given by
dν
x,yα,β(
z
) =
K
α,β(
x, y, z
)
A
α,β(
z
)
dz,
if
x, y
∈
R
∗,
δ
x,
if
y
= 0
,
δ
y,
if
x
= 0.
Here,
δ
xis the Dirac measure at
x
. And,
K
α,β(
x, y, z
)
=
M
α,β(sinh(
|
x
|
) sinh(
|
y
|
) sinh(
|
z
|
))
−2αI
Ix,y×
∫
π0
ρ
θ(
x, y, z
)
×
(
g
θ(
x, y, z
))
α+−β−1sin
2βθdθ,
where
I
x,y=
[
−|
x
| − |
y
|
,
−||
x
| − |
y
||
]
∪
[
||
x
| − |
y
||
,
|
x
|
+
|
y
|
]
ρ
θ(
x, y, z
)
=
1
−
σ
x,y,zθ+
σ
z,x,yθ+
σ
θz,y,x,
∀
z
∈
R
, θ
∈
[0
, π
]
, σ
x,y,zθ=
cosh(x)+cosh(y)−cosh(z) cos(θ)
sinh(x) sinh(y)
,
if
xy
̸
= 0
,
0
,
if
xy
= 0
,
g
θ(
x, y, z
) = 1
−
cosh
2(
x
)
−
cosh
2(
y
)
−
cosh
2(
z
)+2 cosh(
x
) cosh(
y
) cosh(
z
) cos
θ,
t
+=
t,
if
t >
0,
0
,
if
t
≤
0,
and,
M
α,β=
2−2ρΓ(α+1) √
πΓ(α−β)Γ(β+12)
,
if
α > β
,
0
,
if
α
=
β
.
In [2], we have
F
α,β(
T
hf
)(
λ
)
=
ψ
λα,β(
h
)
F
α,β(
f
)(
λ
)
,
(1.3)
F
α,β(Λ
α,βf
)(
λ
)
=
iλ
F
α,β(
f
)(
λ
)
.
(1.4)
For
α
≥
−21, we introduce the normalized Bessel function of first kind
and order
α
[10] defined by:
j
α(
x
) = Γ(
α
+ 1)
∞
∑
n=0
(
−
1)
n(
x/
2)
2nn
!Γ(
n
+
α
+ 1)
,
x
∈
R
.
Moreover, we see that
lim
x→0
j
α(
x
)
−
1
x
2̸
= 0
,
by consequence, there exists
C
1>
0 and
η >
0 satisfying
Lemma 1.2.
Let
α
≥
β
≥
−21, α
̸
=
−21. Then for
|
ν
| ≤
ρ
, there exists a
positive constant
C
2such that
|
1
−
φ
α,βµ+iν(
x
)
| ≥
C
2|
1
−
j
α(
µx
)
|
.
Proof.
(See [4], Lemma 9).
Denote by
L
mp(Λ
α,β)
,
1
< p
≤
2
, m
= 0
,
1
,
2
...
, the class of functions
f
∈
L
pα,β(
R
) that have on
R
generalized derivatives
f
′(
x
)
, f
′′(
x
)
, ...., f
(2m)(
x
)
in the sense of Levi (see [9]) and belong to
L
pα,β(
R
) with Λ
α,βmf
∈
L
pα,β(
R
).
i.e.,
L
mp(Λ
α,β) =
{
f
∈
L
pα,β(
R
)
/
Λ
mα,βf
∈
L
pα,β(
R
)
}
,
where Λ
α,β0f
=
f
, Λ
mα,βf
= Λ
α,β(Λ
α,βm−1f
)
, m
= 0
,
1
,
2
...
.
2.
Main result
In this section we give the main result of this paper. We need first to
define (
ϕ, p
)-Lipschitz Jacobi-Dunkl class.
Denote
N
hby
N
h=
T
h+
T
−h−
2
I,
where
I
is the unit operator in the space
L
pα,β(
R
).
Definition 2.1.
A function
f
∈
L
mp(Λ
α,β) is said to be in (
ϕ, p
)-Lipschitz
Jacobi-Dunkl class, denoted by
Lip
(
ϕ, p, α, β
), if
∥
N
hΛ
mα,βf
∥
p,α,β=
O
(
ϕ
(
h
))
,
as
h
→
0
,
where
m
= 0
,
1
,
2
, ...
and
ϕ
is a continuous increasing function on [0
,
∞
),
satisfying
ϕ
(0) = 0 and
ϕ
(
ts
) =
ϕ
(
t
)
ϕ
(
s
) for all
t, s
∈
[0
,
∞
).
Lemma 2.2.
For
f
∈
L
mp(Λ
α,β)
, then
(∫
R
2
q
λ
qm|
φ
α,βµ
(
h
)
−
1
|
q|F
α,βf
(
λ
)
|
qdσ
(
λ
)
)
1q
≤
K
∥
N
hΛ
mα,βf
∥
p,α,β,
where
1p+
1q= 1
and
m
= 0
,
1
,
2
, ...
Proof.
From (1.4), we have
F
α,β(Λ
mα,βf
)(
λ
) =
i
mλ
mF
α,β(
f
)(
λ
)
,
m
= 0
,
1
,
2
, ....
(2.1)
We use formulas (1.3) and (2.1), we conclude that
F
α,β(
N
hΛ
mα,βf
)(
λ
) =
i
m(
ψ
(α,β)λ
(
h
) +
ψ
(α,β)λ
(
−
h
)
−
2)
λ
mF
α,β
(
f
)(
λ
)
,
Since
ψ
(α,β)λ(
h
) =
φ
α,βµ(
h
) +
i
λ
4(
α
+ 1)
sinh(2
h
)
φ
ψ
(α,β)λ(
−
h
) =
φ
α,βµ(
−
h
)
−
i
λ
4(
α
+ 1)
sinh(2
h
)
φ
α+1,β+1 µ
(
−
h
)
,
and
φ
α,βµis even, then
F
α,β(
N
hΛ
mα,βf
)(
λ
) = 2
i
m(
φ
α,βµ(
h
)
−
1)
λ
mF
α,β(
f
)(
λ
)
.
By formula 1.2, we have the result.
Theorem 2.3.
Let
f
belong to
Lip
(
ϕ, p, α, β
)
. Then
∫
|λ|≥r
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
) =
O
(
ϕ
(
r
−q))
,
as
r
→ ∞
,
where
1p+
1q= 1
and
m
= 0
,
1
,
2
, ...
Proof.
Assume that
f
∈
Lip
(
ϕ, p, α, β
), then we have
∥
N
hΛ
mα,βf
∥
p,α,β=
O
(
ϕ
(
h
))
,
as
h
→
0
.
From Lemma 2.2, we have
∫
R
λ
qm
|
φ
α,βµ
(
h
)
−
1
|
q|F
α,βf
(
λ
)
|
qdσ
(
λ
)
≤
K
q2
q∥
N
hΛ
m α,β
f
∥
q p,α,β
.
By (1.5) and Lemma 1.2, we get:
∫
η 2h≤|λ|≤
η h
λ
qm|
1
−
φ
α,βµ(
h
)
|
q|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≥
C
1qC
q 2∫
η 2h≤|λ|≤
η h
|
µh
|
2qλ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
.
From
2hη≤ |
λ
| ≤
ηhwe have
(
η
2
h
)
2−
ρ
2≤
µ
2≤
(
η
h
)
2−
ρ
2⇒
µ
2h
2≥
η
2
4
−
ρ
2
h
2.
Take
h
≤
3ρη, then we have
µ
2h
2≥
C
3=
C
3(
η
).
So,
∫
η 2h≤|λ|≤
η h
λ
qm|
1
−
φ
α,βµ(
h
)
|
q|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≥
C
1qC
q 2C
q 3∫
η 2h≤|λ|≤η h
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
.
There exists then a positives constants
C
and
K
1such that
∫
η 2h≤|λ|≤
η h
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≤
C
∫
R
λ
qm
|
1
−
φ
α,βµ
(
h
)
|
q|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≤
K
1ϕ
q(
h
) =
K
1ϕ
(
h
q)
.
For all 0
< h <
3ρη. Then we have,
∫
r≤|λ|≤2r
where
K
2=
K
1ϕ
(
η
q2
−q).
Furthermore, we obtain
∫
|λ|≥r
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
=
(∫
r≤|λ|≤2r
+
∫
2r≤|λ|≤4r
+
∫
4r≤|λ|≤8r
+
· ··
)
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≤
K
2ϕ
(
r
−q) +
K
2ϕ
((2
r
)
−q) +
K
2ϕ
((4
r
)
−q) +
· · ·
≤
K
2ϕ
(
r
−q) +
K
2ϕ
(2
−q)
ϕ
(
r
−q) +
K
2ϕ
((2
−q)
2)
ϕ
(
r
−q) +
· · ·
≤
K
2ϕ
(
r
−q)(1 +
ϕ
(2
−q) +
ϕ
((2
−q)
2) +
· · ·
)
.
We have
ϕ
(2
−q)
<
1, then
∫
|λ|≥r
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
)
≤
K
3ϕ
(
r
−q)
,
where
K
3=
K
2(1
−
ϕ
(2
−q))
−1.
Finally, we get
∫
|λ|≥r
λ
qm|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
) =
O
(
ϕ
(
r
−q))
,
as
r
→ ∞
.
Thus, the proof is finished.
Corollary 2.4.
Let
f
∈
L
mp(Λ
α,β)
, and let
∥
N
hΛ
mα,βf
∥
p,α,β=
O
(
ϕ
(
h
))
,
as
h
→
0
.
Then
∫
|λ|≥r
|F
α,β(
f
)(
λ
)
|
qdσ
(
λ
) =
O
(
r
−qmϕ
(
r
−q))
,
as
r
→ ∞
,
where
1p+
1q= 1
and
m
= 0
,
1
,
2
, ...
.
Acknowledgements
The authors would like to thank the referee for his valuable comments
and suggestions.
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