Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES IN q−FOURIER ANALYSIS
LAZHAR DHAOUADI, AHMED FITOUHI AND J. EL KAMEL
Ecole Préparatoire d’Ingénieur
Bizerte, Tunisia.
EMail: lazhardhaouadi@yahoo.fr
volume 7, issue 5, article 171,
2006.
Received 17 April, 2006;
accepted 28 July, 2006.
Communicated by: H.M. Srivastava
Faculté des Sciences de Tunis
1060 Tunis, Tunisia.
EMail: Ahmed.Fitouhi@fst.rnu.tn
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
115-06
Quit
Abstract
In this paper we introduce the q−Bessel Fourier transform, the q−Bessel translation operator and the q−convolution product. We prove that the q−heat semigroup is contractive and we establish the q−analogue of Babenko inequalities
associated to the q−Bessel Fourier transform. With applications and finally we
enunciate a q−Bessel version of the central limit theorem.
Inequalities in q−Fourier
Analysis
2000 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary
26D10.
Key words: Čebyšev functional, Grüss inequality, Bessel, Beta and Zeta function
bounds.
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Contents
1
Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Normalized Hahn-Exton q−Bessel Function . . . . . . . . . .
3
q−Bessel Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
q−Bessel Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . .
5
q−Convolution Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
q−Heat Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
q−Wiener Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
q−Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Title Page
3
5
7
9
12
15
18
25
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
1.
Introduction and Preliminaries
In introducing q−Bessel Fourier transforms, the q−Bessel translation operator
and the q−convolution product we shall use the standard conventional notation
as described in [4]. For further detailed information on q−derivatives, Jackson
q−integrals and basic hypergeometric series we refer the interested reader to
[4], [10], and [8].
The following two propositions will useful for the remainder of the paper.
Inequalities in q−Fourier
Analysis
Proposition 1.1. Consider 0 < q < 1. The series
(w; q)∞1 φ1 (0, w; q; z) =
∞
X
(−1)n q
n=0
n(n−1)
2
(wq n ; q)∞ n
z ,
(q; q)n
defines an entire analytic function in z, w, which is also symmetric in z, w:
(w; q)∞1 φ1 (0, w; q; z) = (z; q)∞1 φ1 (0, z; q; w).
Title Page
Contents
JJ
J
Both sides can be majorized by
|(w; q)∞1 φ1 (0, w; q; z)| ≤ (−|w|; q)∞ (−|z|; q)∞ .
II
I
Go Back
Close
Finally, for all n ∈ N we have
(q 1−n ; q)∞1 φ1 (0, q 1−n ; q; z) = (−z)n q
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
n(n−1)
2
Quit
(q 1+n ; q)∞1 φ1 (0, q 1+n ; q; q n z).
Page 3 of 31
Proof. See [10].
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Now we introduce the following functional spaces:
Rq = {∓q n ,
n
R+
q = {q , n ∈ Z}.
n ∈ Z},
Let Dq , Cq,0 and Cq,b denote the spaces of even smooth functions defined on
Rq continuous at 0, which are respectively with compact support, vanishing at
infinity and bounded. These spaces are equipped with the topology of uniform
convergence, and by Lq,p,v the space of even functions f defined on Rq such
that
Z ∞
p1
p 2v+1
kf kq,p,v =
|f (x)| x
dq x < ∞.
0
We denote by Sq the q−analogue of the Schwartz space of even function f
defined on Rq such that Dqk f is continuous at 0, and for all n ∈ N there is Cn
such that
Cn
|Dqk f (x)| ≤
, ∀k ∈ N, ∀x ∈ R+
q .
(1 + x2 )n
Ar the end of this section we introduce the q−Bessel operator as follows
1 ∆q,v f (x) = 2 f (q −1 x) − (1 + q 2v )f (x) + q 2v f (qx) .
x
Proposition 1.2. Given two functions f and g in Lq,2,v such that
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
∆q,v f, ∆q,v g ∈ Lq,2,v
then
Z
∞
∆q,v f (x)g(x)x
0
2v+1
Z
dq x =
Page 4 of 31
∞
f (x)∆q,v g(x)x
0
Quit
2v+1
dq x.
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
2.
The Normalized Hahn-Exton q−Bessel Function
The normalized Hahn-Exton q−Bessel function of order v is defined as
jv (x, q) =
(q, q)∞ −v (3)
x Jv (x, q) = 1 φ1 (0, q v+1 , q, qx2 ),
(q v+1 , q)∞
<(v) > −1,
(3)
where Jv (·, q) is the Hahn-Exton q−bessel function, (see [12]).
Proposition 2.1. The function
x 7→ jv (λx, q 2 ),
is a solution of the following q−difference equation
∆q,v f (x) = −λ2 f (x)
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
Proof. See [9].
In the following we put
cq,v =
1
(q 2v+2 , q 2 )∞
·
.
1−q
(q 2 , q 2 )∞
Proposition 2.2. Let n, m ∈ Z and n 6= m, then we have
Z ∞
q −2n(v+1)
2
cq,v
jv (q n x, q 2 )jv (q m x, q 2 )x2v+1 dq x =
δnm .
1−q
0
Proof. See [10].
Inequalities in q−Fourier
Analysis
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proposition 2.3.
(−q 2 ; q 2 )∞ (−q 2v+2 ; q 2 )∞
|jv (q , q )| ≤
(q 2v+2 ; q 2 )∞
n
2
(
1
2 +(2v+1)n
qn
if
n ≥ 0,
if
n < 0.
Proof. Use Proposition 1.1.
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
3.
q−Bessel Fourier Transform
The q−Bessel Fourier transform Fq,v is defined as follows
Z ∞
Fq,v (f )(x) = cq,v
f (t)jv (xt, q 2 )t2v+1 dq t.
0
Proposition 3.1. The q−Bessel Fourier transform
Fq,v : Lq,1,v → Cq,0 ,
satisfying
kFq,v (f )kCq,0 ≤ Bq,v kf kq,1,v ,
where
Bq,v =
1 (−q 2 ; q 2 )∞ (−q 2v+2 ; q 2 )∞
.
1−q
(q 2 ; q 2 )∞
Proof. Use Proposition 2.3.
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
Theorem 3.2. Given f ∈ Lq,1,v then we have
2
Fq,v
(f )(x) = f (x),
Inequalities in q−Fourier
Analysis
∀x ∈ R+
q .
If f ∈ Lq,1,v and Fq,v (f ) ∈ Lq,1,v then
II
I
Go Back
Close
Quit
kFq,v (f )kq,2,v = kf kq,2,v .
Page 7 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. Let t, y ∈ R+
q , we put
(
δq,v (t, y) =
It is not hard to see that
Z
1
(1−q)t2v+2
if
t = y,
0
if
t 6= y.
∞
f (t)δq,v (t, y)t2v+1 dq t = f (y).
Inequalities in q−Fourier
Analysis
0
By Proposition 2.2, we can write
Z ∞
2
cq,v
jv (yx, q 2 )jv (tx, q 2 )x2v+1 dq x = δq,v (t, y),
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
∀t, y ∈ R+
q ,
0
Title Page
which leads to the result.
Contents
Corollary 3.3. The transformation
Fq,v : Sq → Sq ,
JJ
J
II
I
Go Back
is an isomorphism, and
−1
Fq,v
= Fq,v .
Proof. The result is deduced from properties of the space Sq .
Close
Quit
Page 8 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
4.
q−Bessel Translation Operator
We introduce the q−Bessel translation operator as follows:
Z ∞
v
Tq,x f (y) = cq,v
Fq,v (f )(t)jv (xt, q 2 )jv (yt, q 2 )t2v+1 dq t,
0
∀x, y ∈ R+
q , ∀f ∈ Lq,1,v .
Proposition 4.1. For any function f ∈ Lq,1,v we have
v
Tq,x
f (y)
=
v
Tq,y
f (x),
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
and
v
Tq,x
f (0) = f (x).
Proposition 4.2. For all x, y ∈ R+
q , we have
v
Tq,x
jv (λy, q 2 ) = jv (λx, q 2 )jv (λy, q 2 ).
Proof. Use Proposition 2.2.
Proposition 4.3. Let f ∈ Lq,1,v then
Z ∞
v
Tq,x f (y) =
f (z)Dv (x, y, z)z 2v+1 dq z,
0
where
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 31
Dv (x, y, z) = c2q,v
Z
0
∞
jv (xt, q 2 )jv (yt, q 2 )jv (zt, q 2 )t2v+1 dq t.
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. Indeed,
v
Tq,x
f (y)
Z ∞
= cq,v
Fq,v (f )(t)jv (xt, q 2 )jv (yt, q 2 )t2v+1 dq t
Z0 ∞ Z ∞
2 2v+1
cq,v
f (z)jv (zt, q )z
dq t jv (xt, q 2 )jv (yt, q 2 )t2v+1 dq t
= cq,v
Z0 ∞
Z ∞0 2
2
2
2 2v+1
=
f (z) cq,v
jv (xt, q )jv (yt, q )jv (zt, q )t
dq t z 2v+1 dq z,
0
0
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
which leads to the result.
Proposition 4.4.
lim Dv (x, y, z) = 0
Title Page
z→∞
and
Contents
(1 − q)
X
q
(2v+2)s
s
Dv (x, y, q ) = 1
s∈Z
Proof. To prove the first relation use Proposition 3.1. The second identity is
v
deduced from Proposition 4.2: if f = 1 then Tq,x
f = 1.
Proposition 4.5. Given f ∈ Sq then
v
Tq,x
f (y)
=
∞
X
n=0
q n(n+1)
y 2n ∆nq,v f (x).
2
2
2v+2
2
(q , q )n (q
, q )n
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. By the use of Proposition 2.1 and the fact that
Z ∞
n
n
∆q,v f (x) = (−1) cq,v
Fq,v (f )(t)t2n jv (xt, q 2 )t2v+1 dq t.
0
Proposition 4.6. If v = − 12 then
q 2(r−m)(k−m)−m 2(r−m)+1
(q
; q)∞1 φ1 (0, q 2(r−m)+1 , q; q 2(k−m)+1 ).
Dv (q , q , q ) =
(1 − q)(q; q)∞
m
r
k
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Proof. Indeed
∆nq,v
Inequalities in q−Fourier
Analysis
n (k+n)(k+n+1)
q −n(n+1) X
2n
−2kn k
2
(−1)k+n q
Λq ,
=
2n
x
k
+
n
q
k=−n
and use Proposition 4.5.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
5.
q−Convolution Product
In harmonic analysis the positivity of the translation operator is crucial. It plays
a central role in establishing some useful results, such as the property of the
convolution product. Thus it is natural to investigate when this property holds
v
for Tq,x
. In the following we put
Qv = {q ∈ [0, 1],
v
Tq,x
is positive for all x ∈ R+
q }.
v
v
Recall that Tq,x
is said to be positive if Tq,x
f ≥ 0 for f ≥ 0.
Proposition 5.1. If v =
− 12
Inequalities in q−Fourier
Analysis
then
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Qv = [0, q0 ],
where q0 is the first zero of the following function:
Title Page
q 7→ 1 φ1 (0, q, q, q).
Contents
v
Proof. The operator Tq,x
is positive if and only if
Dv (x, y, q s ) ≥ 0,
We replace
x
y
JJ
J
∀x, y, q s ∈ R+
q .
by q r , and we can choose r ∈ N, because
Go Back
Close
v
v
Tq,x
f (y) = Tq,y
f (x),
Quit
thus we get
(q 1+2s , q)∞1 φ1 (0, q 1+2s , q, q 1+2r ) =
II
I
∞
X
n=0
Page 12 of 31
Bn (s, r),
∀r, s ∈ N,
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
where
Bn (s, r) =
and
∞
2n
Y
q 2r+i Y
(1 − q 2s+i )
i
1
−
q
i=1
i=2n+2
q 2r+2n+1
2s+2n+1
× (1 − q
)−
,
1 − q 2n+1
∀n ∈ N∗ ,
∞
Y
q 2r+1
2s+i
2s+1
B0 (s, r) =
(1 − q
) (1 − q
)−
,
1−q
i=2
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
which leads to the result.
In the rest of this work we choose q ∈ Qv .
Proposition 5.2. Given f ∈ Lq,1,v then
Z ∞
Z
v
2v+1
Tq,x f (y)y
dq y =
0
Title Page
∞
f (y)y 2v+1 dq y.
0
The q−convolution product of both functions f, g ∈ Lq,1,v is defined by
Z ∞
v
f ∗q g(x) = cq
Tq,x
f (y)g(y)y 2v+1 dq y.
0
Proposition 5.3. Given two functions f, g ∈ Lq,1,v then
f ∗q g ∈ Lq,1,v ,
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 31
and
Fq,v (f ∗q g) = Fq,v (f )Fq,v (g).
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. We have
kf ∗q gkq,1,v ≤ kf kq,1,v kgkq,1,v .
On the other hand
Z
∞
Z
Fq,v (f ∗q g)(λ) =
0
∞
v
f (x)Tq,y
jv (λx, q 2 )x2v+1 dq x
g(y)y 2v+1 dq y
0
= Fq,v (f )(λ) Fq,v (g)(λ).
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
6.
q−Heat Semigroup
The q−heat semigroup is defined by:
v
Pq,t
f (x) = Gv (·, t, q 2 ) ∗q f (x)
Z ∞
v
= cq,v
Tq,x
Gv (y, t, q 2 )f (y)y 2v+1 dq y,
∀f ∈ Lq,1,v .
0
v
Gv (·, t, q 2 ) is the q−Gauss kernel of Pq,t
−2v
q
(−q 2v+2 t, −q −2v /t; q 2 )∞
2 2
G (x, t, q ) =
e −
x ,q .
(−t, −q 2 /t; q 2 )∞
t
v
2
and e(·, q) the q-exponential function defined by
e(z, q) =
∞
X
n=0
zn
1
=
,
(q, q)n
(z; q)∞
|z| < 1.
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
Proposition 6.1. The q−Gauss kernel Gv (·, t, q 2 ) satisfying
Fq,v Gv (·, t, q 2 ) (x) = e(−tx2 , q 2 ),
and
JJ
J
II
I
Go Back
Fq,v
2
2
e(−ty , q ) (x) = Gv (x, t, q 2 ).
Proof. In [5], the Ramanujan identity was proved
q2
2 2
bz,
,
q
,
q
s
X
bz
z
∞,
= 2s
2
2
q
(bq
,
q
)
2
∞
b, z, b , q
s∈Z
∞
Close
Quit
Page 15 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
which implies
Z ∞
X (q 2n+2v+2 )s
e(−ty 2 , q 2 )y 2n y 2v+1 dq y = (1 − q)
(−tq 2s , q 2 )∞
0
s
q −2n−2v
2n+2v+2
2 2
−tq
,− t ,q ,q
∞.
= (1 − q)
q2
2
2n+2v+2
−t, q
,− t ,q
∞
Inequalities in q−Fourier
Analysis
The following identity leads to the result
(a, q 2 )∞ = (a, q 2 )n (q 2n a, q 2 )∞ ,
and
(aq
−2n
2
n −n2 +n
, q )∞ = (−1) q
a
q2
n q2 2
,q
a
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
(a, q 2 )∞ .
n
Proposition 6.2. For any functions f ∈ Sq , we have
v
Pq,t
f (x) = e(t∆q,v , q 2 )f (x).
Proof. Indeed, if
Z ∞
cq,v
Gv (y, t, q 2 )y 2n y 2v+1 dq y = (q 2v+2 , q 2 )n q −n(n+n) tn ,
0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
then
v
Pq,t
f (x)
=
∞
X
n=0
q n(n+1)
(q 2 , q 2 )n (q 2v+2 , q 2 )n
Z
× cq,v
∞
v
2
2n 2v+1
G (y, t, q )y y
dq y ∆nq,v f (x).
0
Inequalities in q−Fourier
Analysis
Theorem 6.3. For f ∈ Lq,p,v and 1 ≤ p < ∞, we have
v
kPq,t
f kq,p,v ≤ kf kq,p,v .
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Proof. If p = 1 then
Title Page
v
kPq,t
f kq,1,v ≤ kGv (·, t, q 2 )kq,1,v kf kq,1,v = kf kq,1,v .
Contents
Now let p > 1 and we consider the following function
JJ
J
v
g : y 7→ Tq,x
Gv (y, t; q 2 ).
Go Back
In addition
v p
Pq,t f ≤ cpq,v
q,p
II
I
Z
∞
Z
|f (y)g(y)| y
0
2v+1
Close
p
∞
dq y
x
2v+1
dq x.
Quit
0
By the use of the Hölder inequality and the fact that kGv (·, t, q 2 )kq,1,v =
the result follows immediately.
1
,
cq,v
Page 17 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
7.
q−Wiener Algebra
For u ∈ Lq,1,v and λ ∈ R+
q , we introduce the following function
x
1
uλ : x 7→ 2v+2 u
.
λ
λ
Proposition 7.1. Given u ∈ Lq,1,v such that
Z ∞
u(x)x2v+1 dq x = 1,
0
then we have
Z
lim
λ→0
∞
f (x)uλ (x)x2v+1 dq x = f (0),
∀f ∈ Cq,b .
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
0
Corollary 7.2. The following function
Gvλ : x 7→ cq,v Gv (x, λ2 , q 2 ),
checks the conditions of the preceding proposition.
Proof. Use Proposition 6.1.
Theorem 7.3. Given f ∈ Lq,1,v ∩ Lq,p,v , 1 ≤ p < ∞ and fλ defined by
Z ∞
fλ (x) = cq
Fq,v (f )(y)e(−λ2 y 2 , q 2 )jv (xy, q 2 )y 2v+1 dq y.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
0
Page 18 of 31
then we have
lim kf − fλ kq,p,v = 0.
λ→0
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. We have
f
∗q Gvλ (x)
Z
∞
= cq,v
Fq,v (f )(t)e(−λ2 t2 , q 2 )jv (tx, q 2 )t2v+1 dq t.
0
In addition, for all ε > 0, there exists a function h ∈ Lq,p,v with compact support
in [q k , q −k ] such that
kf − hkq,p,v < ε,
however
Inequalities in q−Fourier
Analysis
kGvλ ∗q f − f kq,p,v ≤ kGvλ ∗q (f − h)kq,p,v + kGvλ ∗q h − hkq,p,v + kf − hkq,p,v .
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
By Theorem 6.3 we get
kGvλ ∗q (f − h)kq,p,v ≤ kf − hkq,p,v .
Contents
Now, we will prove that
lim kGvλ ∗q h − hkq,p,v = 0.
λ→0
Indeed, by the use of Corollary 7.2 we get
Z 1
lim
|Gvλ ∗q h(x) − h(x)|p x2v+1 dq x = 0.
λ→0
Title Page
0
On the other hand the following function is decreasing on the interval [1, ∞[:
u 7→ u2v+2 Gv (u).
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
If λ < 1, then we deduce that
v
v
v
Tq,q
i Gλ (x) ≤ Tq,q i G(x).
We can use the dominated convergence theorem to prove that
Z ∞
lim
|Gvλ ∗q h(x) − h(x)|p x2v+1 dq x = 0.
λ→0
1
Inequalities in q−Fourier
Analysis
Corollary 7.4. Given f ∈ Lq,1,v then
Z ∞
f (x) = cq,v
Fq,v (f )(y)jv (xy, q 2 )y 2v+1 dq y,
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
∀x ∈ R+
q .
0
Title Page
Proof. The result is deduced by Theorem 7.3 and the following relation
(1 − q)x2v+2 |f (x) − fλ (x)| ≤ kf − fλ kq,1,v
∀x ∈ R+
q .
Now we attempt to study the q−Wiener algebra denoted by
Contents
JJ
J
II
I
Go Back
Close
Aq,v = {f ∈ Lq,1,v ,
Fq,v (f ) ∈ Lq,1,v } .
Proposition 7.5. For 1 ≤ p ≤ ∞, we have
1. Aq,v ⊂ Lq,p,v
and
Aq,v = Lq,p,v .
Quit
Page 20 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
2. Aq,v ⊂ Cq,0
and
Aq,v = Cq,0 .
Proof. 1. Given h ∈ Lq,p,v with compact support, and we put hn = h ∗q Gvqn .
The function hn ∈ Aq,v and by Theorem 7.3 we get
lim kh − hn kq,p,v = 0.
n→∞
2. If f ∈ Cq,0 , then there exist h ∈ Cq,0 with compact support on [q k , q −k ], such
that
kf − hkCq,0 < ε,
and by Corollary 7.4 we prove that
"
lim
n→∞
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
#
sup |h(x) − hn (x)| = 0.
x∈R+
q
Title Page
Contents
Theorem 7.6. For f ∈ Lq,2,v ∩ Lq,1,v , we have
kFq,v (f )kq,2,v = kf kq,2,v .
JJ
J
II
I
Go Back
Close
Proof. We put
fn = f
∗q Gvqn ,
which implies
Fq,v (fn )(t) = e(−q 2n t2 , q 2 )Fq,v (f )(t),
Quit
Page 21 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
by Corollary 7.4 we get
Z
fn (x) = cq
∞
Fq,v (fn )(t)jv (tx, q 2 )t2v+1 dq t.
0
On the other hand
Z ∞
Z
2v+1
f (x)fn (x)x
dq x =
0
∞
Fq,v (f )(x)Fq,v (fn )(x)x2v+1 dq x.
0
Theorem 7.3 implies
Z ∞
lim
Fq,v (f )(x)2 e(−q 2n x2 , q 2 )x2v+1 dq x = kf k2q,2,v .
n→∞
0
The sequence e(−q 2n x2 , q 2 ) is increasing. By the use of the Fatou-Beppo-Levi
theorem we deduce the result.
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
Theorem 7.7.
1. The q−cosine Fourier transform Fq,v possesses an extension
U : Lq,2,v → Lq,2,v .
JJ
J
II
I
Go Back
2. For f ∈ Lq,2,v , we have
Close
kU (f )kq,2,v = kf kq,2,v .
3. The application U is bijective and
U −1 = U.
Quit
Page 22 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. Let the maps
u : Aq,v → Aq,v ,
f 7→ Fq,v (f ).
Theorem 3.2 implies
ku(f )kq,2,v = kf kq,2,v .
The map u is uniformly continuous, with values in complete space Lq,2,v . It has
a prolongation U on Aq,v = Lq,2,v .
Proposition 7.8. Given 1 < p ≤ 2 and p1 + p10 = 1, if f ∈ Lq,p,v , then Fq,v (f ) ∈
Lq,p0 ,v ,
kFq,v (f )kq,p0 ,v ≤ Bp,q,v kf kq,p,v ,
where
( 2 −1)
Bp,q,v = Bq,vp .
Proof. The result is a consequence of Proposition 3.1, Theorem 7.7 and the
Riesz-Thorin theorem, see [13].
As an immediate consequence of Proposition 7.8, we have the following
theorem:
Theorem 7.9. Given 1 < p, p0 , r ≤ 2 and
1
1
1
+ 0 −1= ,
p p
r
if f ∈ Lq,p,v and g ∈ Lq,p0 ,v , then
f ∗q g ∈ Lq,r,v ,
and
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
kf ∗q gkq,r,v ≤ Bq,p,v Bq,p0 ,v Bq,r0 ,v kf kq,p,v kgkq,p0 ,v ,
where
1
1
+ 0 = 1.
r r
Proof. We can write
f ∗q g = Fq,v {Fq,v (f )Fq,v (g)} ,
the use of Proposition 7.8 and the Hölder inequality leads to the result.
Now we are in a position to establish the hypercontractivity of the q−heat
v
semigroup Pq,t
. For more information about this notion, the reader can consult
([1, 2, 3]).
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Proposition 7.10. For f ∈ Lq,p0 ,v and t ∈ R+
q , we have
v
kPq,t
f kq,p,v ≤ Bq,p0 ,v Bq,p1 ,v c(r, q, v)t−
kf kq,p0 ,v ,
1 < p0 < p ≤ 2,
1
1
+
= 1,
p p1
1
1
1
= 0− ,
r
p
p
c(r, q, v) = ke(−x2 , q 2 )kq,r,v .
Proof. The result is deduced by the following relations
Fq,v Gv (·, t, q 2 ) (x) = e(−tx2 , q 2 ),
and
Title Page
Contents
where
and
v+1
r
v+1
kFq,v Gv (·, t, q 2 ) kq,r,v = c(r, q, v)t− r .
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
8.
q−Central Limit Theorem
In this section we study the analogoue of the well known central limit theorem
with the aid of the q−Bessel Fourier transform.
+
For this, we consider the set M+
q of positive and bounded measures on Rq .
The q-cosine Fourier transform of ξ ∈ M+
q is defined by
Z ∞
Fq,v (ξ)(x) =
jv (tx, q 2 )t2v+1 dq ξ(t).
0
The q−convolution product of two measures ξ, ρ ∈ M+
q is given by
Z ∞
v
ξ ∗q ρ(f ) =
Tq,x
f (t)t2v+1 dq ξ(x)dq ρ(t),
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
0
and we have
Fq,v (ξ ∗q ρ) = Fq,v (ξ)Fq,v (ρ).
We begin by showing the following result
Proposition 8.1. For f ∈ Aq,v and ξ ∈ M+
q , we have
Z ∞
Z ∞
f (x)x2v+1 dq ξ(x) = cq,v
Fq,v (f )(x)Fq,v (ξ)(x)x2v+1 dq x.
0
Contents
JJ
J
II
I
Go Back
0
As a direct consequence we may state
Corollary 8.2. Given ξ, ξ 0 ∈ M+
q such that
Fq,v (ξ) = Fq,v (ξ 0 ),
then ξ = ξ 0 .
Title Page
Close
Quit
Page 25 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proof. By Proposition 8.1, we have
Z ∞
Z ∞
2v+1
f (x)x
dq ξ(x) =
f (x)x2v+1 dq ξ 0 (x),
0
∀f ∈ Aq,v .
0
from the assertion (2) of Proposition 7.5, we conclude that ξ = ξ 0 .
Theorem 8.3. Let (ξn )n≥0 be a sequence of probability measures of M+
q such
that
lim Fq,v (ξn )(t) = ψ(t),
n→∞
then there exists ξ ∈ M+
q such that the sequence ξn converges strongly toward
ξ, and
Fq,v (ξ) = ψ.
Proof. We consider the map In defined by
Z ∞
In (u) =
u(x)x2v+1 dq ξn (x),
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
∀f ∈ Cq,0 .
0
By the following inequality
JJ
J
II
I
Go Back
|In (u)| ≤ kukCq,0 ,
and by Proposition 8.1, we get
Z ∞
In (f ) = cq,v
Fq,v (f )(x)Fq,v (ξn )(x)x2v+1 dq x,
Close
Quit
∀f ∈ Aq,v ,
Page 26 of 31
0
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
which implies
∞
Z
Fq,v (f )(x)ψ(x)x2v+1 dq x,
lim In (f ) =
n→∞
∀f ∈ Aq,v .
0
On the other hand, by assertion (2) of Proposition 7.5, and by the use of the
Ascoli theorem (see [11]):
Consider a sequence of equicontinuous linear forms on Cq,0 which converge
on a dense part Aq,v then converge on the entire Cq,0 . We get
Z ∞
lim In (u) =
Fq,v (u)(x)ψ(x)x2v+1 dq x, ∀u ∈ Cq,0 .
n→∞
0
Finally there exist ξ ∈ M+
q such that
Z ∞
Z
2v+1
u(x)x
dq ξn (x) =
lim
n→∞
0
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
∞
u(x)x
2v+1
dq ξ(x),
∀u ∈ Cq,0 .
0
Contents
JJ
J
On the other hand
Fq,v (Aq,v ) = Aq,v ,
and
Z ∞
Inequalities in q−Fourier
Analysis
II
I
Go Back
Z
Fq,v (f )(x)Fq,v (ξ)(x)dq x =
0
∞
Fq,v (f )(x)ψ(x)x2v+1 dq x,
0
which implies
∀f ∈ Aq,v ,
Close
Quit
Page 27 of 31
Fq,v (ξ) = ψ.
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
Proposition 8.4. Given ξ ∈ M+
q , and supposing that
Z ∞
σ=
t2 t2v+1 dq ξ(t) < ∞,
0
then
Fq,v (ξ)(x) = 1 −
q2σ
x2 + o(x2 ).
(q 2 , q 2 )1 (q 2v+2 , q 2 )1
Proof. We write
2 2
jv (tx, q 2 ) = 1 −
q t
x2 + x2 θ(tx)t2 ,
(q 2 , q 2 )1 (q 2v+2 , q 2 )1
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
where
lim θ(x) = 0,
Title Page
x→0
then
Contents
q2σ
Fq,v (ξ)(x) = 1 − 2 2
x2 +
2v+2
2
(q , q )1 (q
, q )1
Z
∞
2
2v+1
t θ(tx)t
dq ξ(t) x2 .
0
JJ
J
II
I
Go Back
Now we are in a position to present the q−central limit theorem.
Theorem 8.5. Let (ξn )n≥0 be a sequence of probability measures of M+
q of
total mass 1, satisfying
Z ∞
lim nσn = σ, where σn =
t2 t2v+1 dq ξn (t),
n→∞
0
Close
Quit
Page 28 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
and
∞
t4 2v+1
t
dq ξn (t),
n→∞
1 + t2
0
then ξn∗n converge strongly toward a measure ξ defined by
2σ
− 2 2 q 2v+2
t2
(q ,q )1 (q
,q 2 )1
dq ξ(x) = cq,v Fq,v e
(x)dq x.
Z
lim ne
σn = 0,
σ
en =
where
Proof. We have
and
Fq,v (ξn∗n ) = (Fq,v (ξn ))n ,
Inequalities in q−Fourier
Analysis
q 2 σn
Fq,v (ξn )(x) = 1 − 2 2
x2 + θn (x)x2 ,
(q , q )1 (q 2v+2 , q 2 )1
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
where
Z
θn (x) =
Title Page
∞
2
2v+1
t θ(tx)t
dq ξn (t).
Contents
0
Consequently
n
(Fq,v (ξn )) (x) = exp n log 1 −
q 2 σn
x2 + θn (x)x2
2
2
2v+2
2
(q , q )1 (q
, q )1
.
JJ
J
II
I
Go Back
Close
By the following inequality
|t2 θ(tx)| ≤ Cx
t4
,
1 + t2
Quit
∀t ∈ R+
q ,
where Cx is some constant, the result follows immediately.
Page 29 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
References
[1] K.I. BABENKO, An inequality in the theory of Fourier integrals, Izv.
Akad. Nauk SSSR, 25 (1961), English transl., Amer. Math. Soc.
[2] W. BECKNER, Inequalities in Fourier analysis, Ann.of Math.,(2), 102
(1975), 159–182.
[3] A. FITOUHI, Inégalité de Babenko et inégalité logarithmique de Sobolev
pour l’opérateur de Bessel, C.R. Acad. Sci. Paris, 305(I) (1987), 877–880.
[4] G. GASPER AND M. RAHMAN, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge University
Press, 1990.
[5] M.E.H. ISMAIL, A simple proof of Ramanujan’s 1 ψ1 sum, Proc. Amer.
Math. Soc., 63 (1977), 185–186.
[6] F.H. JACKSON, On q-Functions and a certain difference operator, Transactions of the Royal Society of London, 46 (1908), 253–281.
[7] F.H. JACKSON, On a q-definite integral, Quarterly Journal of Pure and
Application Mathematics, 41 (1910), 193–203.
[8] T.H. KOORNWINDER, Special functions and q-commuting variables, in
Special Functions, q-Series and Related Topics, M. E. H. Ismail, D. R.
Masson and .Rahman (eds), Fields Institue Communications 14, American
Mathematical Society, 1997, pp. 131–166.
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au
[9] H.T. KOELINK AND R.F. SWARTTOUW, On the zeros of the HahnExton q-Bessel function and associated q-Lommel polynomials, Journal
of Mathematical Analysis and Applications, 186(3) (1994), 690–710.
[10] T.H. KOORNWINDER AND R.F. SWARTTOUW, On q-Analogues of the
Hankel and Fourier transform, Trans. A.M.S., 1992, 333, 445–461.
[11] L. SCHWARTZ, Analye Hilbertienne, Hermann Paris-Collection Méthode, 1979.
[12] R.F. SWARTTOUW, The Hahn-Exton q-Bessel functions, PhD Thesis,
The Technical University of Delft, 1992.
[13] G.O. THORIN, Kungl.Fysiogr.söllsk.i Lund Förh, 8 (1938), 166–170.
Inequalities in q−Fourier
Analysis
Lazhar Dhaouadi, Ahmed Fitouhi
and J. El Kamel
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 31
J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006
http://jipam.vu.edu.au