Gen. Math. Notes, Vol. 28, No. 2, June 2015, pp.9-20
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Generalization of Titchmarsh’s Theorem
for the Jacobi-Dunkl Transform
A. Belkhadir1 and A. Abouelaz2
1,2
Department of Mathematics and Informatics
Faculty of Sciences Aı̈n Chock
University of Hassan II, Casablanca, Morocco
1
E-mail: ak.belkhadir@gmail.com
2
E-mail: ah.abouelaz@gmail.com
(Received: 23-4-15 / Accepted: 29-5-15)
Abstract
In this paper, using a generalized Jacobi-Dunkl translation operator, we
prove a generalization of Titchmarsh’s theorem for functions in the k-JacobiDunkl-Lipschitz class defined by the finite differences of order k ∈ N∗ and
Sobolev spaces associated with the Jacobi-Dunkl operator.
Keywords: Generalized Jacobi-Dunkl translation, Jacobi-Dunkl Lipschitz
class, Jacobi-Dunkl transform, Titchmarsh’s theorem.
1
Introduction
Titchmarsh’s theorem characterizes the set of functions satisfying the CauchyLipschitz condition by means of an asymptotic estimate growth of the norm
of their Fourier transform, namely we have:
Theorem 1.1. [12] Let α ∈ (0, 1) and f ∈ L2 (R) . Then the following
are equivalents:
1.
2.
kf (t + h) − f (t)k = O(hα ) , as h → 0 ;
Z
|fˆ(λ)|2 dλ = O(r−2α ) , as r → +∞ .
|λ|≥r
10
A. Belkhadir et al.
where fˆ is the Fourier transform of f .
A similar result of theorem 1.1 has been established for the Jacobi transform
(see [8], theorem 2.2). Furthermore, a generalization of this result was proved
in the Sobolev spaces associated with Jacobi transform (see [1], theorem 2.1 ).
In this paper, we prove a similar result for Jacobi-Dunkl transform, we con2,k
(associated with Jacobi-Dunkl operator
sider functions in Sobolev spaces Wα,β
(see [5])) belonging to the k-Jacobi-Dunkl-Lipschitz class defined by the finite
difference of order k ∈ N∗ . For this purpose we use the generalized translation
and Jacobi-Dunkl operators.
The paper is organized as follows: in section 2 we recapitulate some results
related to the harmonic analysis associated with the Jacobi-Dunkl operator
Λα,β (see [2, 3, 4, 5, 7]). Section 3 is devoted to the main result (theorem
2,k
3.3). Before, we define the class Lip(δ, 2, α, β) of functions in Wα,β
satisfying
a certain condition correspondent to the generalized Jacobi-Dunkl translation.
Titchmarsh’s theorem for Jacobi-Dunkl transform is given as a corollary of
theorem 3.3.
2
Notations and Preliminaries
In the following, α , β and ρ denote 3 reals such that α ≥ β ≥ − 21 ,
α 6= − 21 and ρ = α + β + 1 .
Notations:
• Aα,β (x) = 2ρ (sinh |x|)2α+1 (cosh |x|)2β+1 .
• dσα,β (λ) =
where,
and
|λ|
8π
p
p
IR\]−ρ,ρ[ (λ)dλ
λ2 − ρ2 |Cα,β ( λ2 − ρ2 )|
Cα,β (µ) =
2ρ−iµ Γ(α + 1)Γ(iµ)
, µ ∈ C \ (iN) .
Γ( 21 (ρ + iµ))Γ( 12 (α − β + 1 + iµ))
IΩ is the characteristic function of Ω .
• Lp (Aα,β ) (resp. Lp (σα,β ) , p ∈]0, +∞[ , the space of measurable functions
g on R such that
Z
||g||Lp (Aα,β ) =
1/p
|g(t)| Aα,β (t)dt
< +∞ .
p
R
Z
(resp. ||g||Lp (σα,β ) =
R
1/p
|g(λ)| dσα,β (λ)
< +∞) .
p
11
Generalization of Titchmarsh’s Theorem...
• D(R) the space of C ∞ -functions on R with compact support.
• S(R) the usual Schwartz space of C ∞ -functions on R rapidly decreasing
together with their derivatives, equipped with the topology of semi-norms
Lm,n , (m, n) ∈ N2 , where
Lm,n (f ) =
sup
x∈R,0≤k≤n
k
d
(1 + x ) k f (x) < +∞.
dx
2 m
• S 1 (R) = {(cosh t)−2ρ f ; f ∈ S(R)} .
The topology of this space is given by the semi-norms L1m,n , (m, n) ∈ N2 ,
where
L1m,n (f )
=
sup
x∈R,0≤k≤n
(cosh t)
−2ρ
k
d
(1 + x ) k f (x) < +∞.
dx
2 m
0
• (S 1 (R)) the topological dual of S 1 (R) .
Now, we introduce the Jacobi-Dunkl Transform and its basic properties:
The Jacobi-Dunkl function with parameters (α, β) , α ≥ β ≥ − 21 , α 6= − 12 ,
is defined by :
(
i d (α,β)
(α,β)
ϕµ (x) , if λ ∈ C \ {0};
ϕµ (x) −
(α,β)
(1)
∀x ∈ R, ψλ (x) =
λ dx
1
, if λ = 0.
(α,β)
with λ2 = µ2 + ρ2 , ρ = α + β + 1 and ϕµ
is the Jacobi function given by:
ρ + iµ ρ − iµ
(α,β)
2
ϕµ (x) = F
(2)
,
; α + 1, −(sinh x) ,
2
2
where F is the Gaussian hypergeometric function given by
F (a, b, c, z) =
∞
X
(a)m (b)m m
z , |z| < 1,
(c)
m!
m
m=0
a, b, z ∈ C and c ∈
/ −N;
(a)0 = 1, (a)m = a(a + 1)...(a + m − 1) . (see [2, 9, 10]).
(α,β)
ψλ
is the unique C ∞ -solution on R of the differentiel-difference equation
Λα,β u = iλu , λ ∈ C;
(3)
u(0) = 1.
where Λα,β is the Jacobi-Dunkl operator given by:
12
A. Belkhadir et al.
A0α,β (x) u(x) − u(−x)
du
Λα,β u(x) =
(x) +
×
; i.e.
dx
Aα,β (x)
2
Λα,β u(x) =
u(x) − u(−x)
du
(x) + [(2α + 1) coth x + (2β + 1) tanh x] ×
.
dx
2
(α,β)
The function ψλ
(α,β)
ψλ
can be written in the form below (See [3]),
(x) + i
(x) = ϕ(α,β)
µ
λ
sinh(2x)ϕ(α+1,β+1)
(x) , ∀x ∈ R ,
µ
4(α + 1)
(4)
where λ2 = µ2 + ρ2 , ρ = α + β + 1.
The Jacobi-Dunkl transform of a function f ∈ L1 (Aα,β ) is defined by :
Z
(α,β)
Fα,β (f )(λ) =
f (x)ψ−λ (x)Aα,β (x)dx, ∀λ ∈ R ;
(5)
R
The inverse Jacobi-Dunkl transform of a function h ∈ L1 (σα,β ) is:
Z
(α,β)
−1
h(λ)ψλ (t)dσα,β (λ) .
Fα,β (h)(t) =
(6)
R
Fα,β is a topological isomorphism from S 1 (R) onto S(R) , and extends uniquely
to a unitary isomorphism from L2 (Aα,β ) onto L2 (σα,β ) . The Plancherel formula
is given by
kf kL2 (Aα,β ) = kFα,β (f )kL2 (σα,β ) .
(7)
For f ∈ S 1 (R) we have the following inversion formula
Z
(α,β)
f (x) =
Fα,β (f )(λ)ψλ (x)dσα,β (λ) , ∀x ∈ R ,
(8)
R
and the relation
Fα,β (Λα,β f )(λ) = iλFα,β (f )(λ) .
(9)
Let f ∈ L2 (Aα,β ) . For all x ∈ R the operator of Jacobi-Dunkl translation
τx is defined by:
Z
α,β
τx f (y) =
f (z)dνx,y
(z) , ∀ y ∈ R .
(10)
R
where
α,β
νx,y
, x, y ∈ R are the signed measures given by

 Kα,β (x, y, z)Aα,β (z)dz , if x, y ∈ R∗ ;
α,β
δx
, if y = 0;
dνx,y
(z) =

δy
, if x = 0.
(11)
13
Generalization of Titchmarsh’s Theorem...
Here, δx is the Dirac measure at x. And
Kα,β (x, y, z) = Mα,β (sinh(|x|) sinh(|y|) sinh(|z|))−2α IIx,y ×
× (gθ (x, y, z))α−β−1
sin2β θdθ.
+
Rπ
0
ρθ (x, y, z)
Ix,y = [−|x| − |y|, −||x| − |y||] ∪ [||x| + |y||, |x| + |y|] ,
θ
θ
θ
ρθ (x, y, z) = 1 − σx,y,z
+ σz,x,y
+ σz,y,x
θ
σx,y,z

 cosh(x) + cosh(y) − cosh(z) cos(θ)
, if xy 6= 0;
=
sinh(x) sinh(y)

0
, if xy = 0.
for all 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+ =
and
Mα,β
t , if t > 0;
0 , if t ≤ 0.

−2ρ
 √ 2 Γ(α + 1)
, if α > β;
=
πΓ(α − β)Γ(β + 21 )

0
, if α = β.
We have
Fα,β (τh f )(λ) = ψλα,β (h).Fα,β (f )(λ) ;
h, λ ∈ R .
Let g ∈ L2 (σα,β ) . Then the distribution Tgσα,β defined by
Z
hTgσα,β , ϕi =
g(λ)ϕ(λ)dσα,β (λ) , ϕ ∈ D(R) ,
(12)
(13)
R
belongs to S 0 (R) .
Let f ∈ L2 (Aα,β ) . Then the distribution Tf Aα,β defined by
Z
hTf Aα,β , ϕi =
f (x)ϕ(x)Aα,β (x)dx , ϕ ∈ S 1 (R) ,
(14)
R
0
belongs to (S 1 (R)) .
Via the correspondance f 7→ Tf Aα,β , we identify L2 (Aα,β ) as a subspace of
0
(S 1 (R)) .
0
The jacobi-dunkl transform of a distribution T ∈ (S 1 (R)) is defined by:
−1
hFα,β (T ), ϕi = hT, Fα,β
(ϕ̌)i , ϕ ∈ S(R) ,
where ϕ̌ is given by ϕ̌(x) = ϕ(−x) .
(15)
14
A. Belkhadir et al.
It is clear that Fα,β (T ) ∈ S 0 (R) .
The jacobi-dunkl transform of a distribution defined by f ∈ L2 (Aα,β ) is
given by the distribution TFα,β (f )σα,β ; i.e.
Fα,β (Tf Aα,β ) = TFα,β (f )σα,β .
(16)
We identify the tempered distribution given by Fα,β (f ) and the function Fα,β (f ) .
0
Let T ∈ (S 1 (R)) and consider the distribution Λα,β T defined by
hΛα,β (T ), ϕi = −hT, Λα,β (ϕ)i , for all ϕ ∈ S 1 (R) .
(17)
(Note that S 1 (R) is unvariant under Λα,β ) .
By using (9) it is easy to see that
Fα,β (Λα,β (T )) = iλFα,β (T ) .
(18)
For f ∈ L2 (Aα,β ) , we define the finite differences of first and higher order
as follows:
∆1h f = ∆h f = τh f + τ−h f − 2f = (τh + τ−h − 2E)f ;
k
∆kh f = ∆h (∆k−1
k = 2, 3, ...;
h )f = (τh + τ−h − 2E) f ,
where E is the unit operator in L2 (Aα,β ) .
Lemma 2.1. The following inequalities are valids for Jacobi functions ϕα,β
µ (h)
(α,β)
1.
|ϕµ
(h)| ≤ 1 ;
2.
|1 − ϕµ
(α,β)
(h)| ≤ h2 λ2 ;
where λ2 = µ2 + ρ2 .
Proof. (See [11], Lemmas 3.1-3.2)
For α ≥
defined by
−1
2
, we introduce the Bessel normalized function of the first kind
jα (z) = Γ(α + 1)
∞
X
n=0
We see that lim
z→0
η > 0 satisfying
(−1)n ( z2 )2n
n!Γ(n + α + 1)
, z ∈ C.
jα (z) − 1
6= 0 , by consequence, there exists c1 > 0 and
z2
|z| ≤ η ⇒
Lemma 2.2. Let α ≥ β ≥
positive constant c2 such that
−1
2
(α,β)
|jα (z) − 1| ≥ c1 |z|2 .
, α 6=
−1
2
. Then for |υ| ≤ ρ , there exists a
|1 − ϕµ+iυ (t)| ≥ c2 |1 − jα (µt)| .
Proof. (See [6], Lemma 9)
(19)
15
Generalization of Titchmarsh’s Theorem...
3
Main Results
2,k
, k ∈ N , the Sobolev space constructed by the operator
We denote by Wα,β
Λα,β ; i.e.
2,k
Wα,β
= f ∈ L2 (Aα,β ); Λjα,β f ∈ L2 (Aα,β ), j = 0, 1, 2, ..., k ;
(20)
where, Λ0α,β f = f, Λ1α,β f = Λα,β f , Λrα,β f = Λα,β (Λr−1
α,β f ), r = 2, 3, ...
2,k
Definition 3.1. Let δ ∈ (0, 1) and k ∈ N . A function f ∈ Wα,β
is said to
be in the k-Jacobi-Dunkl-Lipschitz class, denoted by Lip(δ, 2, k, r) , if
k+1 r ∆ Λα,β f 2
= O(hδ ) , as h −→ 0,
h
L (A
)
α,β
where r = 0, 1, ..., k.
2,k
Lemma 3.2. Let f ∈ Wα,β
, k ∈ N . Then
Z
k+1 r 2
2k+2
∆ Λα,β f 2
=2
λ2r |1 − ϕµ (h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) ,
h
L (A
)
α,β
R
where r = 0, 1, ..., k.
Proof. We have
(α,β)
Fα,β (τh f + τ−h f − 2f )(λ) = (ψλ
(α,β)
(h) + ψλ
(−h) − 2).Fα,β (f )(λ).
λ
(α+1,β+1)
sinh(2h)ϕµ
(h),
4(α + 1)
λ
(α,β)
(α,β)
(α+1,β+1)
ψλ (−h) = ϕµ (−h) − i
sinh(2h)ϕµ
(−h),
4(α + 1)
(α,β)
Since ψλ
(α,β)
and ϕµ
(α,β)
(h) = ϕµ
(h) + i
is even [See (2)]; then:
Fα,β (τh f + τ−h f − 2f )(λ) = 2(ϕ(α,β)
(h) − 1).Fα,β (f )(λ).
µ
and
k+1
Fα,β (∆k+1
(ϕ(α,β)
(h) − 1)k+1 .Fα,β (f )(λ).
µ
h f )(λ) = 2
(21)
From formula (18), we obtain
Fα,β (Λrα,β f )(λ) = (iλ)r Fα,β (f )(λ) .
Using the formulas (21) and (22) we get
r
k+1
Fα,β (∆k+1
(iλ)r .(ϕ(α,β)
(h) − 1)k+1 .Fα,β (f )(λ).
µ
h Λα,β f )(λ) = 2
By the Plancherel formula (7), we have the result.
(22)
16
A. Belkhadir et al.
2,k
, k ∈ N . Then the following are equivalents:
Theorem 3.3. Let f ∈ Wα,β
f ∈ Lip(δ, 2, k, r) ;
Z ∞
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) = O(s−2δ ) , as s → +∞ .
1.
2.
s
Proof. (1) ⇒ (2): Assume that f ∈ Lip(δ, 2, k, r) ; then
k+1 r ∆ Λα,β f 2
= O(hδ ) as h −→ 0.
h
L (A
)
α,β
by lemma 3.2, we have
Z
λ2r |1 − ϕµ (h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) =
R
1
4k+1
r
2
||∆k+1
h Λα,β f ||
= O(h2δ )
η η
, h ] then |µh| ≤ η (recall that λ2 = µ2 + ρ2 ).
If |λ| ∈ [ 2h
We get by (19):
|jα (µh) − 1| ≥ c1 µ2 h2 .
From |λ| ≥
η
we have,
2h
µ2 h2 ≥
η2
− ρ2 h2 ;
4
then we can find an absolute constant c3 = c3 (η, α, β) such that µ2 h2 ≥ c3
(take h < 1) ; thus,
|jα (µh) − 1| ≥ c1 c3 .
this inequality and lemma 2.2 implys that:
|1 − ϕ(α,β)
(h)| ≥ c1 c2 c3 = C
µ
Hence,
1≤
So,
Z
η
η
≤|λ|≤ h
2h
2r
1
C 2k+2
|1 − ϕ(α,β)
(h)|2k+2 .
µ
2
λ |Fα,β (f )(λ)| dσα,β (λ) ≤
Z
1
C 2k+2
η
η
≤|λ|≤ h
2h
λ2r |1 − ϕ(α,β)
(h)|2k+2
µ
×|Fα,β (f )(λ)|2 dσα,β (λ)
≤
Z
1
C 2k+2
2δ
= O(h ).
R
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ)
µ
17
Generalization of Titchmarsh’s Theorem...
Then we have,
Z
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) = O(s−2δ ) ,
as s → +∞.
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) ≤ K1 s−2δ ,
as s → +∞,
s≤|λ|≤2s
Or equivalently
Z
s≤|λ|≤2s
where K1 is some absolute constant . It follows that,
Z
2r
2
λ |Fα,β (f )(λ)| dσα,β (λ) =
|λ|≥s
∞ Z
X
λ2r |Fα,β (f )(λ)|2 dσα,β (λ)
2i s≤|λ|≤2i+1 s
i=0
∞
X
≤ K1
2i s
−2δ
i=0
−2δ
≤ Ks
.
which proves that:
Z
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) = O(s−2δ ) , as s → +∞.
|λ|≥s
(2) ⇒ (1) : Suppose now that
Z
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) = O(s−2δ ) , as s → +∞.
|λ|≥s
we have to show that:
Z
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) = O(h2δ ) , as h → 0.
µ
R
Write:
Z
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) = I1 + I2 ,
µ
R
where:
Z
I1 =
1
|λ|≤ h
Z
I2 =
1
|λ|> h
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) ;
µ
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ).
µ
18
A. Belkhadir et al.
Estimate I1 and I2 . From (1) of lemma 2.1 we can write,
Z
1
k+1
I2 ≤ 4
λ2r |Fα,β (f )(λ)|2 dσα,β (λ) , (s = )
1
h
|λ|> h
= O(h2δ ).
Using the inequalities (1) and (2) of lemma 2.1 we get
Z
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ)
µ
Z
2k+1
λ2r |1 − ϕ(α,β)
(h)|.|Fα,β (f )(λ)|2 dσα,β (λ)
≤ 2
µ
1
|λ|≤ h
Z
2k+1 2
≤ 2
h
λ2r .λ2 |Fα,β (f )(λ)|2 dσα,β (λ).
I1 =
1
|λ|≤ h
1
|λ|≤ h
Consider the function
∞
Z
λ2r |Fα,β (f )(λ)|2 dσα,β (λ).
ψ(s) =
s
An integration by parts gives:
2k+1 2
2
1
h
Z
h
2r
2
2
2k+1 2
λ .λ |Fα,β (f )(λ)| dσα,β (λ) = 2
1
h
Z
h
0
−s2 ψ 0 (s) ds
0
1
1
− 2 ψ( ) + 2
h
h
= 22k+1 h2
2k+2 2
≤ 2
Z
1
h
Z
h
sψ(s)ds.
0
Since ψ(s) = O(s−2δ ) , we get
Z
1
h
Z
sψ(s)ds = O
0
1
h
!
s1−2δ ds
0
= O(h2δ−2 ).
Hence,
2k+1 2
2
Z
h
1
h
λ2r .λ2 |Fα,β (f )(λ)|2 dσα,β (λ) ≤ 22k+2 h2 O(h2δ−2 ).
0
= O(h2δ )
!
1
h
sψ(s)ds
0
19
Generalization of Titchmarsh’s Theorem...
Finally,
Z
λ2r |1 − ϕ(α,β)
(h)|2k+2 |Fα,β (f )(λ)|2 dσα,β (λ) = I1 + I2
µ
R
= O(h2δ ) + O(h2δ )
= O(h2δ )
Which completes the proof of the theorem.
2,k
Corollary 3.4. Let f ∈ Wα,β
such that f ∈ Lip(δ, 2, k, r) . Then:
Z
|Fα,β (f )(λ)|2 dσα,β (λ) = O s−2δ−2r
, as s → +∞ .
|λ|≥s
If we take k = 0 in theorem 3.3, we deduce an analog of Titchmarsh’s
theorem (theorem 1.1) for the Jacobi-Dunkl transform:
Corollary 3.5. Let δ ∈ (0, 1) and f ∈ L2 (Aα,β ) . Then the following are
equivalents:
1.
kτh f + τ−h f − 2f kL2 (Aα,β ) = O(hδ )
Z
2.
, as h → 0 .
|Fα,β (f )(λ)|2 dσα,β (λ) = O(s−2δ )
, as s → +∞ .
|λ|≥s
References
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and application, Afr. Diaspora J. Math, (2004), 24-46.
[3] H.B. Mohamed, The Jacobi-Dunkl transform on R and the convolution
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