Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 7 Issue 1 (2015), Pages 20-37
GENERALIZED q-BESSEL OPERATOR
(COMMUNICATED BY FRANCISCO MARCELLAN)
LAZHAR DHAOUADI
&
MANEL HLEILI
Abstract. In this paper we attempt to build a coherent q-harmonic analysis
attached to a new type of q-difference operator which can be considered as a
generalized of the q-Bessel operator.
1. Introduction
This paper deals with the increasing relevance of q -Bessel Fourier analysis [3, 10,
16]. We introduce a generalized q-Bessel operator of index (α, β), which is a generalization of the well-known q-Bessel operator [10, 3, 4].
This operator satisfy some various identities and admits generalized q-Bessel functions as eigenfunction, in the same way for the q-Bessel functions. We establish the
orthogonality relation and Sonine representation.
Second , we study a generalized q-Bessel transform and we use the work in [4] to
establish inversion formula , Plancherel formula, generalized q-Bessel translation
operator and generalized q-convolution product. Often we use the crucial properties namely the positivity of the q-Bessel translation operator in [9] to prove the
positivity of the generalized q-Bessel translation operator.
As application, we give the Heisenberg uncertainty inequality for functions in Lq,2,ν
space and the Hardy’s inequality which give an information about how a function
and its generalized q-Bessel Fourier transform are linked.
Finally, we study a generalized version of the q-Modified Bessel functions and we
establish some of its properties.
2. The generalized q-Bessel operator
For α, β ∈ R, we put
ν = (β, α),
ν = (α, β),
2000 Mathematics Subject Classification. Primary 33D15,47A05.
Key words and phrases. Generalized q-Bessel function, Generalized q -Bessel Fourier transform, Uncertainty principle, Hardy’s Theorem, Generalized q-Macdonald function.
c
2015
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted Jun 25, 2014. Published August 22, 2014.
20
GENERALIZED q-BESSEL OPERATOR
21
and
ν + 1 = (α + 1, β), |ν| = α + β.
Throughout this paper, we will assume that 0 < q < 1 and α + β > −1. We
refer to [13] for the definitions, notations, properties of the q-shifted factorials, the
Jackson’s q-derivative and the Jackson’s q-integrals.
The q-shifted factorial are defined by
(a; q)0 = 1,
n−1
Y
(1 − aq k ),
(a; q)n =
(a; q)∞ =
k=0
∞
Y
(1 − aq k ),
k=0
and
n
R+
: n ∈ Z}.
q = {q
The q-derivative of a function f is given by
Dq f (x) =
f (x) − f (qx)
(1 − q)x
if
x 6= 0.
and Dq f (0) = f 0 (0) provided f 0 (0) exists. Note that when f is differentiable, at x,
then Dq f (x) tends to f 0 (x) as q tends to 1− .
The q-Jackson integrals from 0 to a and from 0 to ∞ are defined by [15]
Z a
∞
X
f (x)dq x = (1 − q)a
f (aq n )q n ,
0
n=0
∞
Z
f (x)dq x = (1 − q)
0
∞
X
f (q n )q n ,
n=−∞
provided the sums converge absolutely. Note that
Z b
Dq f (x)dq x = f (b) − f (a), ∀a, b ∈ R+
q .
a
The space Lq,p,ν , 1 ≤ p < ∞ denotes the set of functions on R+
q such that
Z ∞
1/p
kf kq,p,ν =
|f (x)|p x2|ν|+1 dq x
< ∞.
0
Similarly Cq,0 is the space of functions defined on R+
q , continuous in 0 and vanishing
at infinity, equipped with the induced topology of uniform convergence such that
kf kq,∞ = sup |f (x)| < ∞,
x∈R+
q
and Cq,b the space of continuous functions at 0 and bounded on R+
q .
The normalized q-Bessel function is given by
∞
X
q n(n+1)
x2n
jα (x, q 2 ) =
(−1)n 2α+2 2
2 , q2 )
(q
,
q
)
(q
n
n
n=0
= 1 φ1 0, q 2α+2 , q 2 ; q 2 x2 .
The q-Bessel operator is defined as follows
∆q,α f (x) =
f (q −1 x) − (1 + q 2α )f (x) + q 2α f (qx)
,
x2
x 6= 0.
22
LAZHAR DHAOUADI
&
MANEL HLEILI
One can see that x 7→ jα (λx, q 2 ), λ ∈ C is eigenfunction for the operator ∆q,α with
−λ2 as eigenvalue .
Let now introduce the following generalized q-Bessel operator:
−1
2α
2β
2α+2β
f (qx)
e q,ν f (x) = ∆
e q,(α,β) f (x) = f (q x) − (q + q )f (x) + q
∆
,
x2
which can be factorized as follows
∗
e q,ν f (x) = ∂q,α
∆
∂q,β ,
where we have put
∂q,β f (x) =
f (q −1 x) − q 2β f (x)
x
f (x) − q 2α+1 f (qx)
.
x
When ν = (α, 0), the last operator is reduced to the q-Bessel operator ∆q,α (see[3,
4, 9]).
∗
∂q,α
f (x) =
Remark 1. We have
e q,ν f (x)
∆
=
=
=
f (q −1 x) − (q 2α + q 2β )f (x) + q 2α+2β f (qx)
x2
−1
2|ν|
f (q x) − (1 + q )f (x) + q 2|ν| f (qx)
+ V (x)f (x)
x2
∆q,|ν| f (x) + V (x)f (x),
where
V (x) =
(1 + q 2|ν| ) − (q 2α + q 2β )
.
x2
In the rest of this paper, we denote by
e
jq,ν (x, q 2 )
= e
jq,(α,β) (x, q 2 )
= x−2β jα−β (q −β x, q 2 ), ν = (α, β).
Proposition 1. The functions e
jq,ν (., q 2 ) and e
jq,ν (., q 2 ) span the space of solutions
of the following q-differential equation
e q,ν f (x) = −f (x).
∆
Proof. We have
e q,ν e
∆
jq,ν (x, q 2 )
q 2β jα−β (q −β−1 x, q 2 ) − (q 2α + q 2β )jα−β (q −β x, q 2 ) + q 2α+2β q −2β jα−β (q −β+1 x, q 2 )
x2
−β−1
2
2(α−β)
jα−β (q
x, q ) − (1 + q
)jα−β (q −β x, q 2 ) + q 2(α−β) jα−β (q −β+1 x, q 2 )
= q 2β x−2β
x2
= −q 2β x−2β q −2β jα−β (q β x, q 2 )
= x−2β
= −jq,(α,β) (x, q 2 ),
and the result follows.
GENERALIZED q-BESSEL OPERATOR
23
Proposition 2. We have
q β+1
xe
jq,ν+1 (x, q 2 ),
1 − q 2(α−β)+2
(1)
q β+1
2
e
x
j
(x,
q
)
=e
jq,ν (x, q 2 ).
q,ν+1
1 − q 2(α−β)+2
(2)
∂q,β e
jq,ν (x, q 2 ) = −
and
∗
∂q,α
Proof. We have
jα−β (q −β−1 x, q 2 ) − jα−β (q −β x, q 2 )
x
= q 2β x−2β (1 − q)Dq [jα−β (q −β−1 x)].
= q 2β x−2β
∂q,β e
jq,ν (x, q 2 )
From the following formula (see [5])
Dq [jα (t, q 2 )] = −
q2
tjα+1 (qt, q 2 ),
(1 − q)(1 − q 2ν+2 )
we obtain
∂q,β e
jq,ν (x, q 2 )
q β+1
x−2β+1 jα+1−β (q −β x, q 2 )
1 − q 2(α−β)+2
q β+1
= −
xe
jq,ν+1 (x, q 2 ).
1 − q 2(α−β)+2
= −
The relation
∗
e q,ν = ∂q,α
∆
∂q,β ,
leads to the second result (2).
Proposition 3. Let f and g be two linearly independent solutions of the following
q-differential equation
e q,ν y(x) = ± λ2 y(x).
∆
Then there exists a constant c(f, g) 6= 0, such that
h
i
x2|ν| f (x)g(qx) − f (qx)g(x) = c(f, g), ∀x ∈ R+
q .
Proof. The q-wronskian of two functions f and g is defined by
h
i
wy (f, g) = (1 − q) ∂q,β f (y)g(y) − f (y)∂q,β g(y) .
The fact that
h
i
h
i
e q,ν f (x)g(x) − f (x)∆
e q,ν g(x) x2|ν|+1 ,
Dq y 7→ y 2|ν|+1 wy (f, g) (x) = ∆
leads to
Z bh
i
e q,ν f (x)g(x) − f (x)∆
e q,ν g(x) x2|ν|+1 dq x = b2|ν|+1 wb (f, g) − a2|ν|+1 wa (f, g),
∆
a
which prove the result.
e q,ν f ∈ Lq,2,ν . Then
Proposition 4. Let f, g ∈ Lq,2,ν such that ∆
e q,ν f, gi = hf, ∆
e q,ν gi,
h∆
if and only if
wx (f, g) = o(x−2|ν|−1 )
as
x ↓ 0.
(3)
24
LAZHAR DHAOUADI
Proof. In fact
Z b
Z
e q,ν f (x)g(x)x2|ν|+1 dq x−
∆
a
&
MANEL HLEILI
b
e q,ν g(x)x2|ν|+1 dq x = b2|ν|+1 wb (f, g)−a2|ν|+1 wa (f, g).
f (x)∆
a
e q,ν f, g ∈ Lq,2,ν we obtain that
Since ∆
Z b
e q,ν f (x)g(x)x2|v|+1 dq x
lim lim
∆
a↓0 b→∞
< ∞.
a
On the other hand f (x) = o(x−|ν|−1 ) and g(x) = o(x−|ν|−1 ) when x → ∞, then we
have
lim b2|ν|+1 wb (f, g) = 0.
b→∞
This implies that
e q,ν f, gi = hf, ∆
e q,ν gi.
lim a2|ν|+1 wa (f, g) = 0 ⇒ h∆
a↓0
The converse is true.
In the rest of this paper, we put
ν = (α, −n),
n ∈ N.
Proposition 5. The function e
jq,ν (x, q 2 ) has the following Sonine integral representation
Z 1
e
jq,ν (x, q 2 ) = x2n
Wν (t, q 2 )jα (q n xt, q 2 )t2α+1 dq t,
0
where
Wν (t, q 2 ) =
(q 2n , q 2 )∞ (q 2α+2 , q 2 )∞ (q 2 t2 , q 2 )∞
.
(q 2 , q 2 )∞ (q 2(α+n)+2 , q 2 )∞ (q 2n t2 , q 2 )∞
(4)
Proof. Using the following identity (see [5])
Z 1 2 2 2
(q 2n , q 2 )∞
(q t , q )∞
2
cq,α+n jα+n (λ, q ) =
cq,α
jα (λt, q 2 )t2α+1 dq t,
2n t2 , q 2 )
(q 2 , q 2 )∞
(q
∞
0
where
cq,α =
1 (q 2α+2 , q 2 )∞
.
1 − q (q 2 , q 2 )∞
The definition of the function e
jq,ν (x, q 2 ) leads to the result.
Proposition 6. The generalized q-Bessel function e
jq,ν (., q 2 ) satisfies the following
estimate
(−q 2 ; q 2 )∞ (−q 2α+2 ; q 2 )∞ (q 2α+2 , q 2 )n
|e
jq,ν (q k , q 2 )| ≤ q 2kn
(−q 2α+2 ; q 2 )n (q 2α+2 , q 2 )∞
2nk
q
if n + k ≥ 0
×
.
2
2
q (n+k) −(n+k)(2α+1)−2n if n + k < 0
Proof. For all n, k ∈ N, we have
e
jq,ν (q k , q 2 ) = q 2kn jα+n (q n+k , q 2 ).
GENERALIZED q-BESSEL OPERATOR
25
Using the following identity (see [4, 3])
|jα+n (q n+k , q 2 )|
≤
(−q 2 ; q 2 )∞ (−q 2(α+n)+2 ; q 2 )∞
(q 2(α+n)+2 , q 2 )∞
1
if k ≥ −n
×
,
(n+k)2 −(2(α+n)+1)(n+k)
q
if k < −n
we obtain the result.
2
e
Proposition 7. Let x ∈ C∗ \R+
q , then the kernel jq,ν (., q ) has the following asymptotic expansion as |x| → ∞
(q 2 x2 , q 2 )∞ (q 2α+2 , q 2 )n
e
jq,ν (x, q 2 ) ∼ x2n 2 2 2
.
(q x , q )n (q 2α+2 , q 2 )∞
2
Proof. Let x ∈ C∗ \R+
q , the function jα (., q ) has the following asymptotic expansion
as |x| → ∞ (see [6])
(x2 q 2 , q 2 )∞
jα (x, q 2 ) ∼ 2α+2 2 .
(q
, q )∞
∗
+
Then for all x ∈ C \Rq , we have
(q 2 x2 , q 2 )∞ (q 2α+2 , q 2 )n
(x2 q 2+2n , q 2 )∞
e
= x2n 2 2 2
jq,ν (x, q 2 ) ∼ x2n 2(α+n)+2 2
,
(q x , q )n (q 2α+2 , q 2 )∞
(q
, q )∞
which achieves the proof.
Definition 1. We define the following delta by
0
δq,ν (x, y) =
1
if x 6= y
if x = y .
(1−q)x2(|ν|+1)
So that for any function f defined on R+
q , we have
Z ∞
f (y)δq,ν (x, y)y 2|ν|+1 dq y = f (x).
0
Proposition 8. The following orthogonality holds relation
Z ∞
e
c2q,ν
jq,ν (tx, q 2 )e
jq,ν (ty, q 2 )t2|ν|+1 dq t = δq,ν (x, y),
0
where
cq,ν =
q n(α+n)
(q 2α+2 , q 2 )∞
.
2
(1 − q) (q , q 2 )∞ (q 2α+2 , q 2 )n
Proof. ∀x, y ∈ R+
q , we have
Z ∞
e
jq,ν (tx, q 2 )e
jq,ν (ty, q 2 )t2|ν|+1 dq t =
0
=
(xy)2n
Z
(5)
∞
jα+n (xq n t, q 2 )jα+n (yq n t, q 2 )t2(α+n)+1 dq t
0
Z ∞
2n −2n(α+n)
(xy) q
jα+n (xu, q 2 )jα+n (yu, q 2 )u2(α+n)+1 dq u.
0
Using the following formula (see [3])
Z ∞
c2q,α
jα (xu, q 2 )jα (yu, q 2 )u2α+1 dq u = δq,α (x, y).
0
Then the result follows.
26
LAZHAR DHAOUADI
&
MANEL HLEILI
3. Generalized q-Bessel Fourier transform
Definition 2. The generalized q-Bessel Fourier transform Fq,ν is defined as follows
Z ∞
Fq,ν f (x) = cq,ν
f (t) e
jq,ν (tx, q 2 ) t2|ν|+1 dq t,
(6)
0
where cq,ν is given by (5).
Proposition 9. The generalized q-Bessel Fourier transform
Fq,ν : Lq,1;ν → Cq,0 ,
satisfies
kFq,ν f kq,∞ ≤ Bq,ν kf kν,1,q ,
where
Bq,ν =
q n(α+n+2k) (−q 2 , q 2 )∞ (−q 2α+2 , q 2 )∞
.
(1 − q)
(q 2 , q 2 )∞ (−q 2α+2 , q 2 )n
Proof. Use Proposition 6.
Theorem 1.
(1) Let f be a function in the Lν,p,q space where p ≥ 1 then
2
Fq,ν
f = f.
(7)
(2) If f ∈ Lq,1,ν with Fq,ν f ∈ Lq,1,ν then
kFq,ν f kq,2,ν = kf kq,2,ν .
(3) Let f be a function in the Lq,1,ν ∩ Lq,p,ν , where p > 2 then
kFq,ν f kq,2,ν = kf kq,2,ν .
(4) Let f be a function in the Lq,2,ν then
kFq,ν f kq,2,ν = kf kq,2,ν .
(5) Let 1 ≤ p ≤ 2. If f ∈ Lq,p,ν then f ∈ Lq,p,ν .
2
−1
p
kf kq,p,ν ,
kFq,ν f kq,p,ν ≤ Bq,ν
(8)
where the numbers p and p above are conjugate exponents
1
1
=1− .
p
p
Proof. The following proof is identical to the proof of Theorems 1,2 and 3 in [4]. Proposition 10. Let f ∈ Lq,2,ν then
e q,ν f (ξ) = −ξ 2 Fq,ν f (ξ),
Fq,ν ∆
∀ξ ∈ R+
q ,
(9)
if and only if
wx (f, ψξ ) = o(x−2|ν|−1 )
as
x ↓ 0,
∀ξ ∈ R+
q .
(10)
In particular this is true if we have
∂q,β f (x) = O(x−|ν| )
as
x ↓ 0.
Proof. Indeed we have (9) if and only if
e q,ν f, ψξ i = hf, ∆
e q,ν ψξ i.
h∆
By Proposition (4) this is equivalent to (10).
GENERALIZED q-BESSEL OPERATOR
27
The q-Schwartz space Sq,ν denote the set of functions f defined on R+
q such that
c
n,k
e kq,ν f (x)| ≤
|∆
, ∀n, k ∈ N, ∀x ∈ R+
q .
1 + x2n
For some constant cn,k > 0 and
e kq,ν f (x) = O(x−|ν| ), as x ↓ 0.
∂q,β ∆
Corollary 1. The generalized q-Bessel transform
Fq,ν : Sq,ν → Sq,ν
define an isomorphism.
3.1. Generalized q-Bessel Translation Operator. We introduce the generalized q-Bessel translation operator associated via the generalized q-Bessel transform
as follows:
Z ∞
ν
Tq,x
f (y) = cq,ν
Fq,ν f (t) e
jq,ν (yt, q 2 )e
jq,ν (xt, q 2 )t2|ν|+1 dq t, ∀x, y ∈ R+
q .
0
Proposition 11. For any function f ∈ Lq,1,ν , we have
ν
ν
Tq,x
f (y) = Tq,y
f (x).
and
ν
Tq,x
f (0) = f (x).
ν
f exists and we have
Theorem 2. Let f ∈ Lq,p,ν then Tq,x
Z ∞
ν
Tq,x
f (y) =
f (z)Dq,ν (x, y, z)z 2|ν|+1 dq z,
0
where
Dq,ν (x, y, z)
= cq,ν
2
Z
∞
0
(xyz)2n
=
e
jq,ν (xs, q 2 ) e
jq,ν (ys, q 2 ) e
jq,ν (zs, q 2 )s2|ν|+1 dq s
Z ∞
2
cq,α+n
jα+n (xs, q 2 ) jα+n (ys, q 2 ) jα+n (zs, q 2 )s2(α+2n)+1 dq s.
0
ν
in the following form
Proof. We write the operator Tq,x
Z ∞
ν
Tq,x
f (y) = cq,ν
Fq,ν f (z) e
jq,ν (yz, q 2 ) e
jq,ν (xz, q 2 )z 2|ν|+1 dq z
0
Z ∞
Z ∞
2 2|ν|+1
e
= cq,ν
cq,ν
f (t)jq,ν (tz, q )t
dq t e
jq,ν (yz, q 2 ) e
jq,ν (xz, q 2 )z 2|ν|+1 dq z
0
0
Z ∞
Z ∞
2
2 e
2 e
2 2|ν|+1
e
=
f (t) cq,ν
jq,ν (yz, q ) jq,ν (xz, q )jq,ν (tz, q )z
dq z t2|ν|+1 dq t
0
0
Z ∞
=
f (t)Dq,ν (x, y, t)t2|ν|+1 dq t.
0
The computation is justified by the Fubuni’s theorem
Z ∞ Z ∞
|f (t)||e
jq,ν (tz, q 2 )|t2|ν|+1 dq t |e
jq,ν (yz, q 2 ) e
jq,ν (xz, q 2 )|z 2|ν|+1 dq z
0
0
1/p
2 p 2|ν|+1
e
dq t
|e
jq,ν (yz, q 2 ) e
jq,ν (xz, q 2 )|z 2|ν|+1 dq z
≤ kf kq,p,ν
|jq,ν (tz, q )| t
0
0
Z ∞
1
2
e
|e
jq,ν (yz, q 2 ) e
jq,ν (xz, q 2 )|z 2(|ν|+1)(1− p )−1 dq z < ∞,
≤ kf kq,p,ν kjq,ν (., q )kq,p,ν
Z
∞
Z
∞
0
28
LAZHAR DHAOUADI
&
MANEL HLEILI
the result follows.
ν
Recall that the generalized q-Bessel translation operator Tq,x
is said to be positive
if it satisfies :
If f ≥ 0
then
ν
Tq,x
f ≥ 0, ∀ x ∈ R+
q .
ν
Obviously, the positivity of the generalized q-Bessel translation operator Tq,x
is
related to the positivity of the kernel Dq,ν (x, y, t).
Let us denote by Qq,ν the domain of positivity of the generalized q-Bessel translation
operator given by:
Qq,ν = {q ∈]0, 1[, if f ≥ 0
ν
then Tq,x
f ≥ 0, ∀ x ∈ R+
q }.
Lemma 1. We have
Dq,ν (x, y, t) ≥ 0,
∀x, y, t ∈ R+
q .
Proof. From Lemma 5 in [9], we have when
Z ∞
2
jν (zx, q 2 )jν (zy, q 2 )jν (zt, q 2 )z 2ν+1 dq z ≥ 0,
Dν,q (x, y, t) = cq,ν
0
that
Dν+µ,q (x, y, t) = c2q,ν+µ
∞
Z
jν+µ (zx, q 2 )jν+µ (zy, q 2 )jν+µ (zt, q 2 )z 2(ν+α)+1 dq z ≥ 0,
0
where
0 < µ < α < 1.
Put α = 2µ, we obtain
Dν+µ,q (x, y, t) = c2q,ν+µ
∞
Z
jν+µ (zx, q 2 )jν+µ (zy, q 2 )jν+µ (zt, q 2 )z 2(|ν|+2µ)+1 dq z ≥ 0.
0
Then for all k ∈ N, 0 < µ < 1, we have
Z ∞
Dν+kµ,q (x, y, t) = c2q,ν+kµ
jν+kµ (zx, q 2 )jν+kµ (zy, q 2 )jν+kµ (zt, q 2 )z 2(|ν|+2kµ)+1 dq z ≥ 0.
0
For kµ = n and the definition of the kernel Dq,ν (x, y, t) lead to the result.
3.2. Generalized q-Convolution Product.
Definition 3. The Generalized q-convolution product is defined by
f ∗q g = Fq,ν [Fq,ν f × Fq,ν g].
Theorem 3. let 1 ≤ p, r, s such that
1 1
1
+ −1= .
p r
s
Given two functions f ∈ Lq,p,ν and g ∈ Lq,r,ν then f ∗q g exists and we have
Z ∞
ν
f ∗q g(x) = cq,ν
Tq,x
f (y)g(y)y 2|ν|+1 dq y,
0
and
f ∗q g
Fq,ν [f ∗q g]
∈
Lq,s,ν ,
= Fq,ν f × Fq,ν g.
GENERALIZED q-BESSEL OPERATOR
29
If s ≥ 2,
kf ∗q gkq,s,ν ≤ Bq,ν kf kq,p,ν kgkq,r,ν .
Proof. We have
f ∗q g(x)
= Fq,ν [Fq,ν f × Fq,ν g](x)
Z ∞
= cq,ν
Fq,ν f (y) Fq,ν g(y) e
jq,ν (yx, q 2 )y 2|ν|+1 dq y
0
Z ∞
Z ∞
2 2|ν|+1
e
jq,ν (yx, q 2 )y 2|ν|+1 dq y
= cq,ν
Fq,ν f (y) cq,ν
g(t) jq,ν (ty, q )t
dq t e
0
0
Z ∞
Z ∞
2 e
2 2|ν|+1
e
=
cq,ν cq,ν
Fq,ν f (y) jq,ν (ty, q )jq,ν (yx, q )y
dq y g(t)t2|ν|+1 dq t
0
0
Z ∞
ν
= cq,ν
Tq,x
f (t) g(t) t2|ν|+1 dq t.
0
The computation is justified by the Fubuni’s Theorem
Z ∞
Z ∞
2
2|ν|+1
e
|g(t) jq,ν (ty, q )|t
|Fq,ν f (y)|
dq t |e
jq,ν (yx, q 2 )|y 2|ν|+1 dq y
0
0
Z
≤ kgkq,r,ν
∞
∞
Z
|e
jq,ν (ty, q 2 )|r t2|ν|+1 dq t
|Fq,ν f (y)|
0
1/r
|e
jq,ν (yx, q 2 )|y 2|ν|+1 dq y
0
∞
Z
2
≤ kgkq,r,ν ke
jq,ν (ty, q )kq,r,ν
h
i
2|ν|+2
|Fq,ν f (y)| |e
jq,ν (yx, q 2 )|y − r
y 2|ν|+1 dq y
0
≤ kgkq,r,ν
ke
jq,ν (ty, q 2 )kq,r,ν kFq,ν f kq,p,ν
Z
∞
p
|e
jq,ν (ty, q 2 )|p y 2(|ν|+1)(1− r )−1
1/p
< ∞.
0
From (8), we deduce that
Fq,ν f ∈ Lq,p,ν and Fq,ν g ∈ Lq,r,ν .
Hence, using the Hölder inequality and the fact that
1 1
1
+ = ,
p r
s
we conclude that
Fq,ν f × Fq,ν g ∈ Lq,s,ν .
gives
f ∗q g = Fq,ν [Fq,ν f × Fq,ν g] ∈ Lq,s,ν .
From the inversion formula (7), we obtain
Fq,ν [f ∗q g] = Fq,ν f × Fq,ν g.
Suppose that s ≥ 2, so 1 ≤ s ≤ 2 and we can write
kf ∗q gkq,s,ν
=
kFq,ν [Fq,ν f × Fq,ν g]kq,s,ν
≤
s
Bq,ν
kFq,ν f kq,p,ν kFq,ν gkq,r,ν
≤
p
s
r
Bq,ν
Bq,ν
Bq,ν
kf kq,p,ν kgkq,r,ν
≤
Bq,ν kf kq,p,ν kgkq,r,ν .
2
2
−1
−1
2
−1
2
−1
30
LAZHAR DHAOUADI
&
MANEL HLEILI
4. Uncertainty principle
In the survey articles by Folland and Sitaram [12] and by Cowling and Price [2] , one
find various uncertainty principles in the literature. In this section, the Heisenberg
uncertainty inequality is established for functions in Lq,2,ν .
Proposition 12. If h∂q,β f, gi exists and
lim a2|ν|+1 f (q −1 a)g(a) − ε2|ν|+1 f (q −1 ε)g(ε) = 0,
a→∞
ε→0
∗
then f, ∂q,α
g exists and we have
∗
h∂q,β f, gi = −q 2β f, ∂q,α
g .
Proof. Let ε ∈ R+
q . The following computation
Z a
∂q,β f (x)g(x)x2|ν|+1 dq x
ε
Z a
f (q −1 x) − q 2β f (x)
=
g(x)x2|ν|+1 dq x
x
ε
Z a
Z a
f (x)
f (q −1 x)
g(x)x2|ν|+1 dq x − q 2β
g(x)x2|ν|+1 dq x
=
x
x
ε
ε
Z q−1 a
Z a
f (x)
f (x)
2|ν|+1
2|ν|+1
2β
=q
g(qx)x
dq x − q
g(x)x2|ν|+1 dq x
x
x
−1
q ε
ε
Z a
Z a
f (x)
f (x)
2|ν|+1
2|ν|+1
2β
=q
g(qx)x
dq x − q
g(x)x2|ν|+1 dq x + a2|ν|+1 f (q −1 a)g(a)
x
x
ε
ε
−ε2|ν|+1 f (q −1 ε)g(ε)
Z a
g(x) − q 2α+1 g(qx) 2|ν|+1
2β
= −q
x
dq x + a2|ν|+1 f (q −1 a)g(a) − ε2|ν|+1 f (q −1 ε)g(ε)
f (x)
x
ε
Z a
∗
= −q 2β
f (x)∂q,α
g(x)x2|ν|+1 dq x + a2|ν|+1 f (q −1 a)g(a) − ε2|ν|+1 f (q −1 ε)g(ε),
ε
leads to the result.
2
Corollary 2. If f ∈ Lq,2,ν such that x Fq,ν f ∈ Lq,2,ν and
∂q,β f (x) = O(x−|ν| )
as
x ↓ 0.
Then ∂q,β f ∈ Lq,2,ν and we have
k∂q,β f k2 = q β kxFq,ν f k2 .
Proof. In fact
2
q 2β kxFq,ν f k2
= q 2β Fq,ν f, x2 Fq,ν f
D
E
e q,ν f
= −q 2β Fq,ν f, Fq,ν ∆
D
E
2
2 e
= −q 2β Fq,ν
f, Fq,ν
∆q,ν f
D
E
e q,ν f
= −q 2β f, ∆
∗
= −q 2β f, ∂q,α
∂q,β f
2
= h∂q,β f, ∂q,β f i = k∂q,β f k2 ,
GENERALIZED q-BESSEL OPERATOR
which prove the result.
31
Theorem 4. Assume that f belongs to the space Lq,2,ν such that
xf, x2 Fq,ν f ∈ Lq,2,ν
and
∂q,β f (x) = O(x−|ν| )
as
x ↓ 0.
Then the generalized q-Bessel transform satisfies the following uncertainty principle
2
kf k2 ≤ kq,ν kxf k2 kxFq,ν f k2 ,
where
kq,ν
β √
q + q × q α+1
=
.
1 − q 2(|ν|+1)
Proof. In fact
∗
∂q,α
xf = f (x) − q 2α+2 f (qx),
x∂q,β f = f (q −1 x) − q 2β f (x).
We introduce the following operator
Λq f (x) = f (qx),
then
hΛq f, gi = q −2(|ν|+1) f, Λ−1
q g .
So
1
1−
2β ∗
q ∂q,α xf (x) − q 2α+2 Λq x∂q,β f (x) = f (x).
q 2(|ν|+1)
Assume that xf and x2 Fq,ν f belong to the space Lq,2,ν , then we have
hf, f i = −
1
1 − q 2(|ν|+1)
hxf, ∂q,β f i −
q −2β
∂q,β f, xΛ−1
q f .
2(|ν|+1)
1−q
Note that
hxf, ∂q,β f i and ∂q,β f, xΛ−1
exist
q f
and
lim ε2|ν|+2 f (q −1 ε)f (ε) = 0.
ε→0
By Cauchy-Schwartz inequality, we get
hf, f i ≤
1
1 − q 2(|ν|+1)
On the other hand
kxf k2 k∂q,β f k2 +
q −2β
k∂q,β f k2 xΛ−1
q f 2.
1 − q 2(|ν|+1)
−1 xΛq f = √q × q |ν|+1 kxf k .
2
2
Corollary 2 gives the result.
32
LAZHAR DHAOUADI
&
MANEL HLEILI
5. Hardy’s theorem
One of the famous formulations of the uncertainty principle is stated by the socalled Hardy’s theorem [14], and many interesting results about this theorem was
proved in the last years [18, 7, 11, 1]. In this section, we give Hardy’s theorem for
the generalized q-Bessel Fourier transform which its proof are the same as those in
[4].
Theorem 5. Suppose f ∈ Lq,1,ν satisfying the following behaviour
1
2
|f (x)| ≤ Ce− 2 x ,
1
∀x ∈ R+
q ,
2
|Fq,ν f (x)| ≤ Ce− 2 x ,
∀x ∈ R,
where C is a positive constant. Then there exists A ∈ R such that
1 2
f (z) = Acq,ν Fq,ν e− 2 x (z), ∀z ∈ C,
where cq,ν is given by (5).
Corollary 3. Suppose f ∈ Lq,1,ν satisfying the following behaviour
2
|f (x)| ≤ Ce−px ,
∀x ∈ R+
q ,
2
|Fq,ν f (x)| ≤ Ce−σx ,
∀x ∈ R,
where C, p, σ are positive constants with p σ = 41 . We suppose that there exists
1
2
a ∈ R+
q such that a p = 2 . Then there exists A ∈ R such that
2
f (z) = Acq,ν Fq,ν e−σt (z), ∀z ∈ C,
where cq,ν is given by (5).
Corollary 4. Suppose f ∈ Lq,1,ν satisfying the following behaviour
2
|f (x)| ≤ Ce−px ,
∀x ∈ R+
q ,
2
|Fq,ν f (x)| ≤ Ce−σx ,
where C, p, σ are positive constants with pσ >
1
2
a ∈ R+
q such that a p = 2 . Then f ≡ 0.
∀x ∈ R,
1
4.
We suppose that there exists
6. Generalized q-Macdonald function
Definition 4. The generalized modified q-Bessel functions is defined by
a
Iq,ν
(x, q 2 ) = e
jq,ν (iax, q 2 ), i2 = −1, a > 0.
We put
a
γq,ν
(x, q 2 ) = e
jq,ν (ax, q 2 ),
a
a
πq,ν
(x, q 2 ) = γq,ν
(ix, q 2 ).
GENERALIZED q-BESSEL OPERATOR
33
The integral representation of Macdonald function in [20, p. 434] suggests that to
define the generalized q-Macdonald function as follows:
−1
Z ∞
t2
a
e
jq,ν (tx, q 2 ) t2|ν|+1 dq t
Kq,ν
(x, q 2 ) = cq,ν
1+ 2
a
0
−1
Z ∞
t2
2n
= x cq,ν
jα+n (q n tx, q 2 ) t2α+1 dq t,
1+ 2
a
0
where cq,ν is given by (5).
a
Theorem 6. The previous generalized q-Macdonald function Kq,ν
is Lq,1,ν and we
have
−1
x2
a
, ∀x ∈ R+
(11)
Fq,ν (Kq,ν
)(x) = 1 + 2
q .
a
−1
x2
Proof. Since 1 + 2
∈ Lq,p,ν , the inversion formula of the generalized q-Bessel
a
transform leads to the result.
a
a
(λx, q 2 ) are two
(λx, q 2 ) and x → Kq,ν
Proposition 13. The functions x → Iq,ν
linearly independent solutions of the following equation
e q,ν f (x) = λ2 f (x).
∆
Proof. In fact we have
"
#
e q,ν
∆
a
1 − 2 Kq,ν
(x, q 2 )
a
Z
= cq,ν
0
∞
t2
1+ 2
a
−1 "
(12)
#
e q,ν
∆
jq,ν (tx, q 2 )t2|ν|+1 dq t = 0.
1− 2 e
a
Note that
"
#
e q,ν
∆
t2
1− 2 e
jq,ν (tx, q 2 ) = 1 + 2 e
jq,ν (tx, q 2 ).
a
a
a
a
The function Kq,ν
∈ Lq,1,ν but Iq,ν
∈
/ Lq,1,ν . Hence, we conclude that they provide
two linearly independent solutions.
|ν| +
Lemma 2. Let λ ∈ C such that λ ∈
/ R+
q ∪ q Rq , then we have
lim q 2|ν|k
k→∞
e
jq,−ν (q −k−|ν| λ, q 2 )
e
jq,ν (q −k λ, q 2 )
= (−1)n q −n(n−1) λ−2n
(q 2(α+n)+2 , q 2 )∞ (q 2n+2|ν| λ−2 , q 2 )∞ (q 2|ν| λ−2 , q 2 )n (q 2−2|ν| λ2 , q 2 )∞
. (13)
(q −2n λ−2 , q 2 )∞ (q 2+2n λ2 , q 2 )∞
(q −2(α+n)+2 , q 2 )∞
Proof. Let x ∈ C∗ \R+
q , then we have the following asymptotic expansion
(x2 q 2+2n , q 2 )∞
e
jq,ν (x, q 2 ) ∼ x2n 2(α+n)+2 2 , |x| → ∞.
(q
, q )∞
If k → ∞, we have
(q 2−2k+2n λ2 , q 2 )∞ (q 2 , q 2 )∞
e
, ∀λ ∈
/ R+
jq,ν (q −k λ, q 2 ) ∼ q −2kn λ2n 2α+2 2
q .
(q
, q )∞ (q 2(α+n)+2 , q 2 )∞
34
LAZHAR DHAOUADI
&
MANEL HLEILI
On the other hand
(q 2−2k q 2n λ2 , q 2 )∞ = (−1)k q −k(k−1) q 2kn λ2k (q −2n λ−2 , q 2 )k (q 2+2n λ2 , q 2 )∞ .
Hence when n → ∞
(−1)k q −k(k−1) λ2n λ2k (q −2n λ−2 , q 2 )k (q 2+2n λ2 , q 2 )∞
e
jq,ν (q −k λ, q 2 ) ∼
,
(q 2(α+n)+2 , q 2 )∞
∀λ ∈
/ R+
q ,
and for all λ ∈
/ q |ν| R+
q , we have
(−1)k+n q −n(n−1) q −k(k−1) q −2|ν|k λ2k (q 2n+2|ν| λ−2 , q 2 )k (q 2|ν| λ−2 , q 2 )n (q 2−2|ν| λ2 , q 2 )∞
e
jq,−ν (q −k−|ν| λ, q 2 ) ∼
.
(q −2(α+n)+2 , q 2 )∞
This implies
q 2|ν|k
e
jq,−ν (q −k−|ν| λ, q 2 )
e
jq,ν (q −k λ, q 2 )
= (−1)n q −n(n−1) λ−2n
(q 2(α+n)+2 , q 2 )∞ (q 2n+2|ν| λ−2 , q 2 )k (q 2|ν| λ−2 , q 2 )n (q 2−2|ν| λ2 , q 2 )∞
.
(q −2n λ−2 , q 2 )k (q 2+2n λ2 , q 2 )∞
(q −2(α+n)+2 , q 2 )∞
Hence when k → ∞ we obtain the result.
Proposition 14. We have
a
a
a
Kq,ν
(x, q 2 ) = σνa πq,ν
(x, q 2 ) − θνa Iq,ν
(x, q 2 ) ,
(14)
where
θνa
=
a
(q −k , q 2 )
πq,ν
a (q −k , q 2 )
k→∞ Iq,ν
=
a−2α q −n(n−1)
lim
(q 2(α+n)+2 , q 2 )∞ (−q 2n+2|ν| a−2 , q 2 )∞ (−q 2|ν| a−2 , q 2 )n (−q 2−2|ν| a2 , q 2 )∞
,
(−q −2n a−2 , q 2 )∞ (−q 2+2n a2 , q 2 )∞
(q −2(α+n)+2 , q 2 )∞
and
σνa =






(q 2 , q 2 )∞
(q 2|ν| , q 2 )∞
−1
Z ∞



c
t2

 − q,ν
1+ 2
t2|ν|+1 dq t
θνa 0
a
if
|ν| ≥ 0
.
if
− 1 < |ν| < 0
a
a
(λx, q 2 ) and x → πq,ν
(λx, q 2 ) are two linearly indeProof. The functions x → Iq,ν
pendent solutions of (12). Then there exist two constants θνa and σνa such that (14)
hold true. Now we can write
"
#
a
πq,ν
(q −k , q 2 )
a
−k 2
a
a
a
Kq,ν (q , q ) = σν
− θν Iq,ν
(q −k , q 2 ).
a (q −k , q 2 )
Iq,ν
On the other hand
a
lim Iq,ν
(q −k , q 2 ) = ∞.
k→∞
Using Theorem 6, we have
a
lim Kq,ν
(q −k , q 2 ) = 0.
k→∞
GENERALIZED q-BESSEL OPERATOR
35
Then it is necessary that
"
lim
k→∞
#
a
πq,ν
(q −k , q 2 )
a
− θν = 0.
a (q −k , q 2 )
Iq,ν
Formula (13) with λ = ia leads to the result. To estimate σνa , we consider two
cases:
• If |ν| > 0, we obtain
Z ∞
a
2|ν| a
2
e
σν = lim x Kq,ν (x, q ) = cq,ν
jq,ν (t, q 2 )t2|ν|−1 dq t.
x→0
0
Using an identity established in [16], with (θ = α + n, µ = 0, λ = 1 − α, m → ∞).
We conclude that
(q 2 , q 2 )∞
σνa = 2|ν| 2 .
(q , q )∞
• If −1 < |ν| < 0, we see that
σνa
1
cq,ν
a
= − a lim Kq,ν
(x, q 2 ) = − a
θν x→0
θν
Z
∞
0
−1
t2
t2|ν|+1 dq t.
1+ 2
a
Corollary 5. As direct consequence, we have
a
a
c(Iq,ν
, Kq,ν
) = σνa (q −2|ν| − 1).
a
(., q 2 ) satisfies the
Proposition 15. The generalized q-Macdonald function Kq,ν
following properties
a. For all x ∈ R+
q we have
a
∂q,β Kq,ν
(x, q 2 ) = −
q 1−n
a
x Kα+1,n
(x, q 2 ).
1 − q 2(α+n)+2
b. For all x ∈ R+
q we have
a
Kq,ν
∈ Lq,2,ν .
a
c. If f ∈ Lq,1,ν and if h(x) = Kq,ν
∗q g(x) then
∆q,ν
1 − 2 h(x) = f (x).
a
d.There exist c, σ > 0 such that
2
a
|Kq,ν
(q −k , q 2 )| < σck q k ,
and
a
Kq,ν
(q −k , q 2 )
= 0.
a (q −k+1 , q 2 )
k→∞ Kq,ν
lim
Proof. c.) From Theorem 6 and Theorem 1, we see that the generalized q-Macdonald
function belongs to Lq,2,ν .
b.) By Theorem 3, we see that h ∈ Lq,1,ν and we have
−1
x2
Fq,ν h(x) = 1 + 2
Fq,ν g(x).
a
36
LAZHAR DHAOUADI
&
MANEL HLEILI
By (7) we have
Z
∞
h(x) = cq,ν
0
So
"
#
e q,ν
∆
1 − 2 h(x)
a
−1
t2
e
Fq,ν g(t) 1 + 2
jq,ν (tx, q 2 )t2|ν|+1 dq t,
a
Z
=
∞
cq,ν
0
Z
=
cq,ν
#
−1 "
e q,ν
t2
∆
Fq,ν g(t) 1 + 2
1− 2 e
jq,ν (tx, q 2 )t2|ν|+1 dq t
a
a
∞
Fq,ν g(t)e
jq,ν (tx, q 2 )t2|ν|+1 dq t = g(t).
0
d.) Let f be a solution of the q-difference equation
e q,ν f (x) = [W (x) − λ]f (x), ∀x ∈ R+ .
∆
q
(15)
From Remark 1, we have
e q,ν f (x) = ∆q,|ν| f (x) + V (x)f (x),
∆
then the last equation (15) is equivalent to:
∆q,|ν| f (x)
=
[W (x) − V (x) − λ]f (x)
=
[R(x) − λ]f (x),
where
R(x) = W (x) − V (x).
From b.) the generalized q-Macdonald function belongs to Lq,2,ν and we apply
Theorem 4 in [6] with W (x) = 0 and λ = −a2 . This give the result.
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GENERALIZED q-BESSEL OPERATOR
37
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