Volume 2008, Article ID 465909,13pages doi:10.1155/2008/465909
Research Article
On Some Inequalities of Uncertainty Principles
Type in Quantum Calculus
Ahmed Fitouhi,1N ´eji Bettaibi,2 Rym H. Bettaieb,2and Wafa Binous3
1Facult´e des Sciences de Tunis, Universit´e de Tunis El Manar, 1060 Tunis, Tunisia 2Institut Pr´eparatoire aux ´Etudes d’Ing´enieur de Monastir, Universit´e de Monastir,
5000 Monastir, Tunisia
3Institut de Biotechnologie, Universit´e de Jendouba, 9000 B´eja, Tunisia
Correspondence should be addressed to N´eji Bettaibi,[email protected]
Received 19 July 2007; Revised 31 January 2008; Accepted 14 April 2008
Recommended by Wolfgang Castell
The aim of this paper is to generalize theq-Heisenberg uncertainty principles studied by Bettaibi et al.2007, to state local uncertainty principles for theq-Fourier-cosine, theq-Fourier-sine, and theq-Bessel-Fourier transforms, then to provide an inequality of Heisenberg-Weyl-type for the q-Bessel-Fourier transform.
Copyrightq2008 Ahmed Fitouhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The uncertainty principle is a metatheorem in harmonic analysis that asserts, with the use of some inequalities, that a function and its Fourier transform cannot be sharply localized. We refer to the survey article by Folland and Sitaram1and the book of Havin and J ¨oricke 2for various classical uncertainty principles of different nature which may be found in the
literature.
In3, the authors gaveq-analogues of the Heisenberg uncertainty principle for the q-Fourier-cosine and theq-Fourier-sine transforms. One of the aims of this paper is to provide a generalization of their work next to state local uncertainty principles for variousq-Fourier transforms.
q-Bessel-Fourier transforms. Then, we give a Heisenberg-Weyl-type inequality for some q-Bessel-Fourier transform.
2. Notations and preliminaries
Throughout this paper, we assumeq∈0,1. We recall some usual notions and notations used in theq-theorysee4,5. We refer to the book by Gasper and Rahman4for the definitions, notations, and properties of theq-shifted factorials and theq-hypergeometric functions.
We writeRq{±qn:n∈Z},Rq,{qn:n∈Z}, and
xq 1−qx
1−q, x∈C, nq! q;qn
1−qn, n∈N. 2.1
Theq-derivative of a functionfis given by
Dqf
x fx1−−qxfqx ifx /0, 2.2
Dqf0 limk→∞Dqfqk, provided that the limit exists.
Theq-Jackson integrals from 0 toaand from 0 to∞, of a functionf, aresee6 a
0
fxdqx 1−qa ∞
n0
faqnqn,
∞
0
fxdqx 1−q ∞
n−∞
fqnqn, 2.3
provided that the sums converge absolutely.
Theq-Jackson integral in a generic intervala, bis given bysee6 b
a
fxdqx
b
0
fxdqx−
a
0
fxdqx. 2.4
Theq-integration by parts rule is given, for suitable functionsfandg,by
b
a
gxDqfxdqxfbgb−faga−
b
a
fqxDqgxdqx. 2.5
Jacksonsee6defined aq-analogue of the Gamma function by
Γqx q;q∞ qx;q
∞
1−q1−x,
x /0,−1,−2, . . . . 2.6
The third Jacksonq-Bessel functionsee7,8is
Jν
z;q2 z ν
1−q2νΓ
q2ν11
ϕ1
and theq-trigonometric functionsq-cosine andq-sineare defined bysee9
cosx;q2 Γq21/2 q1q−11/2x
1/2J
−1/2
1−q q x;q
2
∞ n0
−1nqnn−1 x2n
2nq! ,
sinx;q2 Γq21/2 1q−11/2x
1/2J 1/2
1−q q x;q
2
∞ n0
−1nqnn−1 x2n1
2n1q! .
2.8
They verify
Dqcos
x;q2−1 qsin
qx;q2, Dqsin
x;q2cosx;q2. 2.9
We need the following spaces and norms.
iS∗qRqis the space of even functionsfonRqsatisfying
∀n, m∈N, Pn,m,qf sup x∈Rq; 0≤k≤n
1x2mDkqfx<∞. 2.10
iiLn
qRq,, x2ν1dqx,n≥1,ν≥ −1/2, is the set of all functions defined onRq,such that
f n,ν,q ∞
0
fxnx2ν1d qx
1/n
<∞. 2.11
iiiLn
qRq, LnqRq,, dqx,n≥1, and · n,q · n,−1/2,q.
ivL∞qRq,is the set of all bounded functions onRq,. We write f ∞,qsupx∈Rq,|fx|.
3. Generalization of the Heisenberg uncertainty principle
Theq-Fourier-cosine and theq-Fourier-sine transforms are defined assee8,9
Fqfx cq
∞
0
ftcosxt;q2dqt, qFfx cq
∞
0
ftsinxt;q2dqt, 3.1
where
cq
1q−11/2
Γq21/2 . 3.2
Lettingq ↑ 1 subject to the conditionLog1−q/Logq ∈ Zgives, at least formally, the classical Fourier transformssee 3,10. In the remainder of the present paper, we assume
It was shown in8,9that we have the following result.
Proposition 3.1. 1Forf∈L1
qRq,, one hasFqf∈L∞qRq,and
Fqf∞,q≤ 1q−11/2
Γq21/2q;q2∞
f 1,q. 3.3
2Fqis an isomorphism ofL2qRq,(resp., S∗,qRq) onto itself. Moreover, one hasF−q1 Fq and the following Plancherel formula:
Fqf 2,q f 2,q, f∈L2qRq,. 3.4
Similarly, it was shown in3,8that theq-Fourier-sine transform verifies the following properties.
Proposition 3.2. 1Forf∈L1
qRq,, one hasqFf∈L∞qRq,and
qFf ∞,q≤
1q−11/2
Γq21/2q;q2∞
f 1,q. 3.5
2qFis an isomorphism ofL2qRq,onto itself; its inverse is given byqF−1 1/q2qF. One has the following Plancherel formula:
qFf 2,qq f 2,q, f∈L2qRq,. 3.6
Let us now state the following useful density result.
Proposition 3.3. For alln≥1,S∗,qRqis dense inLnqRq,.
Proof. Let n ≥ 1 and f ∈ Ln
qRq,. For p ∈ N, put fp f·χqp,q−p, where χqp,q−p is the
characteristic function ofqp, q−p.
It is clear that for allp ∈N,fp ∈S∗,qRqand|f−fp|n ≤ |f|n. So, the Lebesgue theorem implies thatfppconverges tofinLnqRq,.
Remark 3.4. Using the density ofS∗,qRqinLnqRq, n ≥ 1, one can see that the
q-Fourier-cosineresp.,q-Fourier-sinetransform has a unique continuous extension onLn
qRq,, that will also be denoted asFq resp.,qF. We have the followingq-analogue of the Hausdorff-Young inequality.
Theorem 3.5. Letn∈1,2(resp.,n1) andmn/n−1(resp.,m∞) be the dual exponent of n. For allfinLn
qRq,, the functionsFqfandqFfbelong toLmqRq,, and one has
where
C1
1q−11/2
Γq21/2q;q2∞
1−2n−1/n
, C2
1q−11/2
Γq21/2q;q2∞
1−2n−1/n
q2n−1/n. 3.8
Proof. The result is a direct consequence of11, Theorem 1.3.4, page 35, and Propositions3.1
and3.2, by takingS∗,qRqas a set of simple functions.
The following lemma gives relations between the two Fourier q-trigonometric trans-forms.
Lemma 3.6. 1Forf ∈L2
qRq,such thatDqf∈L2qRq,, one has
qF
Dqf
λ −λ qFqf
λ q
, λ∈Rq,. 3.9
2Additionally, if limn→∞fqn 0, then
Fq
Dqf
λ λ
q2qFfλ, λ∈Rq,. 3.10
Proof. The same steps as in the proof of3, Lemma 2; theq-integration by parts rule and the fact that
∞
0
ftdqtn→lim∞
q−n
qn ftdq
t 3.11
give the result.
In 3, the authors proved the following q-analogues of the Heisenberg uncertainty principle.
Theorem 3.7. Letfbe inL2
qRq,such thatDqfis inL2qRq,. Then,
tf 2,q λFqf 2,q≥ q q3/21 f
2
2,q. 3.12
In addition, if limn→∞fqn 0, one has
tf 2,q λqFf 2,q≥ q q−3/21 f
2
2,q. 3.13
Now, we are in a position to generalizeTheorem 3.7. One obvious way to generalize it is to replace theL2
qnorms byLnq norms. This is the purpose of the following result. Theorem 3.8. For 1≤n≤2 andf∈L2
qRq,, one has
f 2
2,q≤C1 xf n,q λFqf n,q, 3.14
f 2
where
C1q−11/n1q−n1/nC2, C2q−1
1q−n1/nC1, 3.16
withC1andC2being given by3.8.
Proof. The casen2 has been dealt with inTheorem 3.7. Now, assume 1≤n <2 and letmbe the dual exponent ofn. Letf ∈S∗,qRqsuch that limt→0ft 0. From the relation
Dq
fft Dqftft fqtDqft, 3.17 theq-integration by parts rule, and the H ¨older inequality, we have, sincet|ft|2tends to 0 ast tends to∞inRq,,
1 q
∞
0
ft2
dqt
∞
0
tDqfftdqt
≤
∞
0
tDqftftdqt
∞
0
tfqtDqftdqt
≤
∞
0
tftn dqt
1/n∞
0
Dqftmdqt
1/m
∞
0
tfqtn dqt
1/n∞
0
Dqft m
dqt
1/m .
3.18
However, the change of variableuqtgives
∞
0
tfqtn dqt
1/n
q−n1/n
∞
0
tftn dqt
1/n . 3.19 So, 1 q ∞ 0
ft2
dqt≤1q−n1/n tf n,q Dqf m,q. 3.20
On the other hand, we haveDqf Fq
Fq
Dqf
q−2
qFqF
Dqf
sinceDqfis in L2qRq,. Then, by usingLemma 3.6and theq-analogue of the Hausdorff-Young inequality, we
obtain
Dqfm,q≤C1Fq
Dqfn,q C1
q2λqFfn,q,
Dqfm,q≤q−2C2qF
Dqfn,qq−2C2
λqFqf
λ
q
n,q
q−21/nC
2λFqfn,q. 3.21 Thus,
f 2 2,q≤q−1
1q−n1/nC1 tf n,qλqFfn,q, 3.22
f 2
2,q≤q−11/n
1q−n1/nC2 tf n,qλFqfn,q. Now, letf∈L2
qRq,; it is easy to see that for allp∈N,fpfχqp,q−p∈S∗,qRq, limt→0fpt 0,
andfppconverges tofinL2qRq,. Moreover, if the right-hand side of3.14 resp.,3.15is
finite, then the functionstf andλFqf resp., λqFfare in LnqRq,,and they are limits
in Ln
4. Local uncertainty principles
In the literature, the first classical local inequalities were obtained by Farissee12in 1978, and they were generalized by Pricesee 13,14 in 1983 and 1987. In this section, we will generalize Price’s results by giving theirq-analogues.
4.1. Local uncertainty principles for theq-Fourier trigonometric transforms
Theorem 4.1. If 0< a <1/2, there is a constantK Ka, qsuch that for all bounded subsetEof Rq,and allf∈L2qRq,, one has
E
Fqfλ2
dqλ≤K|E|2axaf22,q. 4.1
Here, |E| 0∞χExdqxand K cq/
1−2aq1−2a/2a4a1/1−2a2, where cq 1q−11/2/Γ
q21/2q;q2∞.
Proof. Forr >0, letχr χ0,rbe the characteristic function of0, randχr 1−χr. Then, forr >0, we have, sincef·χr∈L1qRq,,
E
Fqfλ2
dqλ
1/2
FqfχE2,q≤Fqf·χrχE2,qFqf·χrχE2,q
≤ |E|1/2F
qf·χr∞,qFqf·χr2,q,
4.2
and by the use of the H ¨older inequality, we obtain
Fqf·χr∞,q≤cqf·χr1,q
cqx−aχr·xaf1,q≤cqx−aχr2,qxaf2,q≤
cq
1−2aq
r1/2−axaf2,q. 4.3
On the other hand, sincef ∈L2
qRq,, we havef·χr∈L2qRq,, and by the Plancherel formula,
we get
Fqf·χr2,qf·χr2,qx−aχr.xaf2,q≤x−aχr∞,qxaf2,q≤r−axaf2,q. 4.4
So,
E
Fqfλ2
dqλ
1/2
≤
⎛ ⎜ ⎝ cq
1−2aq
|E|1/2r1/2−ar−a
⎞ ⎟
⎠xaf2,q. 4.5
Corollary 4.2. For 0< a <1/2 andb >0, there is a constantKa,bsuch that for allf ∈L2qRq,, one
has
f ab
2,q ≤Ka,bxafb2,qλbFqfa2,q. 4.6
Proof. Forr > 0, putEr 0, r ∩ Rq, andEr r,∞ ∩ Rq,. It is easy to see thatEr is a bounded subset ofRq,and|Er| ≤r.
Then, from the Plancherel formula andTheorem 4.1, we have
f 2
2,qFqf22,q
Er
Fqf2λd
qλ
Er
Fqf2λd
qλ
≤Kr2axaf22,qr−2bλbFqf22,q.
4.7
Choosing r > 0 so as to minimize the right-hand side of the inequality, we obtain f 2 2,q ≤ Ka,b xaf b2,q λbFqf a2,q2/ab, withKa,b a/bb/ab b/aa/ab
ab/2
Kb/2, andKis
the constant given inTheorem 4.1.
In the same way, one can prove the following local uncertainty principle for the q-Fourier-sine transform.
Theorem 4.3. If 0< a <1/2, there is a constantKKa, qsuch that for all bounded subsetEof Rq,and allf∈L2qRq,, one has
E
qFfλ2
dqλ≤K|E|2axaf22,q, 4.8
whereK cq/
1−2aq1−2a/2qa
4a
12qa/1−2a2.
Corollary 4.4. For 0< a <1/2 andb >0, there is a constantKa,bsuch that for allf ∈L2
qRq,, one
has
f ab
2,q ≤Ka,b xaf 2b,q λbqFf a2,q, 4.9
withKa,b a/bb/ab b/aa/abab/2Kb/2q−ab.
Proof. The same steps ofCorollary 4.2give the result.
Theorem 4.5. Ifa >1/2, there is a constantK1K1a, qsuch that for all bounded subsetEofRq,
andf∈L2
qRq,, one has
E
Fqfλ2d
qλ≤K1|E| f 22,q−1/axaf
1/a
2,q, 4.10
E
Fqfλ2
dqλ≤K1|E| f 22,q−1/axaf
1/a
The proof of this result needs the following lemmas.
Lemma 4.6. Supposea >1/2, then for allf∈L2
qRq,,such thatxaf∈L2qRq,,
f 2 1,q≤K2
f 2
2,qxaf
2 2,q
, 4.12
whereK2K2a, q 1−qq2a, q2a,−q,−q2a−1;q2a∞/q, q2a−1,−q2a,−1;q2a∞.
Proof. From15, Example 1, and the H ¨older inequality, we have
f 2 1,q
∞
0
1x2a1/2|fx|1x2a−1/2dqx
2
≤K2 f 22,q xaf 22,q, 4.13
whereK2
∞
0 1x2a
−1
dqx 1−qq2a, q2a,−q,−q2a−1;q2a∞/q, q2a−1,−q2a,−1;q2a∞.
Lemma 4.7. Supposea >1/2, then for allf∈L2
qRq,, such thatxaf∈L2qRq,, one has
f 1,q≤K3 f 21,q−1/2axaf
1/2a
2,q , 4.14
whereK3K3a, q 2aK22aq−q1/2a−1 1/2
.
Proof. Fors∈Rq,,define the functionfsbyfsx fsx, x∈Rq,.
We have fs 1,qs−1 f 1,q, xafs 22,qs−2a−1 xaf 22,q. Replacement offbyfsinLemma 4.6gives
f 2
1,q≤K2s f 22,qs−2a1 xaf 22,q. 4.15
Now, for allr >0,putαr Logr/Logq−ELogr/Logq.We haves r/qαr∈R q,
andr≤s < r/q.Then, for allr >0,
f 2 1,q≤K2
r
q f
2
2,qr−2a1xaf
2 2,q
. 4.16
The right-hand side of this inequality is minimized by choosing
r 2a−11/2aq1/2a f −2,q1/axaf12/a,q. 4.17
When this is done, we obtain the result.
Proof ofTheorem 4.5. Since the proofs of the two statements are similar, it is sufficient to prove 4.11.
LetEbe a bounded subset ofRq,. When the right-hand side of the inequality4.11is
finite,Lemma 4.6implies thatf ∈ L1
Proposition 3.1,Lemma 4.7, and the fact that
E
|Fqfλ|2dqλ≤ |E| Fqf 2∞,q, 4.18
we obtain the result withK1 1q−1/Γ2q21/2q;q
4
∞K23.
Remark 4.8. By the same technique as in the proof of Corollary 4.2, we can show that
Theorem 4.5leads to inequalities4.6and4.9with some different constants.
4.2. Local uncertainty principles for theq-Bessel-Fourier transform
Theq-Bessel-Fourier transform is definedsee16forf∈L1
qRq,, x2ν1dqxby
Fν,qfλ cν,q
∞
0
fxjν
λx;q2x2ν1dqx, 4.19
where
jν
z;q21−q2νΓq2ν1
1−qq−1z−νJ
ν
1−qq−1z;q2 4.20
is the normalized third Jacksonq-Bessel function, and
cν,q
1q−1−ν
Γq2ν1. 4.21
It was shown in10that forν≥ −1/2, we have the following result.
Theorem 4.9. 1Forf∈L1
qRq,, x2ν1dqx, one hasFν,qf∈L∞q Rq,and
Fν,qf∞,q≤ cν,q q;q22
∞
f 1,ν,q. 4.22
2Fν,q is an isomorphism ofL2qRq,, x2ν1dqxonto itself,F−ν,q1 q4ν2Fν,q, and one has the following Plancherel formula:
∀f∈L2qRq,, x2ν1dqx
, Fν,q2,ν,qq2ν1 f 2,ν,q. 4.23
The following result states a local uncertainty principle for theq-Bessel-Fourier transform.
Theorem 4.10. Forν≥ −1/2 and 0< a < ν1, there is a constantKa,νKa, ν, qsuch that for all f∈L2qRq,, x2ν1dqxand all bounded subsetEofRq,, one has
E
Fν,qfλ2
λ2ν1d
qλ≤Ka,ν|E|νa/ν1xaf22,ν,q. 4.24
Here,|E|ν
∞
0 χExx2ν1dqx,cν,qcν,q/q;q2 2
∞, and
Ka,ν
⎛ ⎜
⎝ cν,q
2ν2−2aq
⎞ ⎟ ⎠
2a/ν1
aq2ν1
ν1−a
1−a/ν1 q2ν1
aq2ν1
ν1−a
−a/ν12 .
Proof. Letν≥ −1/2, 0< a < ν1,f∈L2
qRq,, x2ν1dqx, and letEbe a bounded subset ofRq,.
Forr >0, we have, sincef.χr ∈L1qRq,, x2ν1dqx,
E
Fν,qfλ2
λ2ν1dqλ
1/2
Fν,qfχE2,ν,q
≤Fν,qf·χrχE2,ν,qFν,qf·χrχE2,ν,q
≤ |E|1/2
ν Fν,qf·χr∞,qFν,qf·χr2,ν,q.
4.26
However, by the use of the H ¨older inequality, we obtain
Fν,q
f·χr∞,q≤cν,qf·χr1,q
cqx−aχr.xaf1,ν,q
≤cν,qx−aχr2,ν,qxaf2,ν,q.
4.27
Now, ifkis the integer such thatqk≤r < qk−1, we get, sincea < ν1,
x−aχr22,ν,q
∞
0
x−2aχ
rxx2ν1dqx
qk
0
x2ν1−2ad qx
q2kν1−a 2ν2−2aq ≤
r2ν1−a
2ν2−2aq. 4.28
Then,
Fν,qf·χr∞,q≤ cν,q
2ν2−2aqr
ν1−axaf
2,ν,q. 4.29
On the other hand, sincef ∈L2
qRq,, x2ν1dqx, we havef.χr ∈L2qRq,, x2ν1dqx,and by the Plancherel formula4.23, we obtain
Fν,q
f·χr2,ν,qq2ν1f·χr2,ν,qq2ν1x−aχr·xaf2,ν,q
≤q2ν1x−aχ
r∞,qxaf2,q≤q2ν1r−axaf2,ν,q.
4.30
So,
E
Fν,qfλ2
λ2ν1dqλ
1/2
≤
⎛ ⎜
⎝ cν,q
2ν2−2aq|E|
1/2
ν rν1−aq2ν1r−a
⎞ ⎟
⎠xaf2,ν,q.
4.31
Theorem 4.11. Forν ≥ −1/2 anda > ν1, there exists a constantKa,ν such that for all bounded subsetEofRq,and allfinLq2Rq,, x2ν1dqx, one has
E
Fν,qfλ2
λ2ν1dqλ≤Ka,ν|E| f
21−ν1/a
2,ν,q xaf
2ν1/a
2,ν,q . 4.32
Proof. Sincea > ν1, the same steps as in the proof ofTheorem 4.5and the relation4.22give the result with
Ka,νq
2a, q2a,−q2ν2,−q2a−ν−1;q2a ∞ q2ν2, q2a−ν−1,−q2a,−1;q2a
∞ cν,q,
cν,q 1−q
cν,q q;q22
∞
2
a ν1−1
ν1/a a a−ν−1
q−2ν1a−ν−1/a.
4.33
Corollary 4.12. Forν≥ −1/2 anda, b >0, there is a constantKa,b,ν Ka, b, ν, qsuch that for all f∈L2
qRq,, x2ν1dqx, one has f ab
2,ν,q ≤Ka,b,ν xaf b2,ν,q λbFν,qf a2,ν,q, 4.34 with Ka,b,ν ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b a
a/ab
a b
b/abab/2
Ka,νb/2
q−2ν1ab
2ν2qab/2ν1 ifa < ν1,
Ka,ν 2ν2q
ab/2ν2
q−4ν2 b
ν1
ν1/νb1
b ν1
−b/νb1aνb1/2ν1
ifa > ν1,
4.35 whereKa,ν(resp.,Ka,ν) is the constant given inTheorem 4.10(resp.,Theorem 4.11).
Proof. Forr >0, we putEr 0, r ∩Rq,andEr r,∞ ∩ Rq,.
We haveEr is a bounded subset ofRq,and|Er|ν ≤r2ν2/2ν2q.Then, the Plancherel formula4.23and Theorems4.10and4.11lead to
q4ν2 f 22,ν,qFν,qf22,ν,q
Er
Fν,qf2λλ2ν1d
qλ
Er
Fν,qf2λλ2ν1d
qλ ≤ ⎧ ⎪ ⎨ ⎪ ⎩
Ka,νErνa/ν1xaf22,ν,qr−2bλbFν,qf22,ν,q ifa < ν1,
Ka,ν Er f 22,ν,qa−ν−1/axaf22,ν,qν1/ar−2bλbFν,qf22,ν,q ifa > ν1,
≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Ka,ν 2ν2a/ν1
q
r2axaf2
2,ν,qr−2bλbFν,qf
2
2,ν,q ifa < ν1,
Ka,ν r
2ν2
2ν2q
f 2a−ν−1/a
2,ν,q xaf
2ν1/a
2,ν,q r−2bλbFν,qf
2
Remark that whena b 1, we obtain a Heisenberg-Weyl-type inequality for the q-Bessel-Fourier transform.
Corollary 4.13. Forν≥ −1/2, ν /0, one has for allf∈L2
qRq,, x2ν1dqx,
f 2
2,ν,q≤K1,1,ν xf 2,ν,qλFν,qf2,ν,q. 4.37
Acknowledgment
The authors would like to thank the reviewers for their helpful remarks and constructive criticism.
References
1 G. B. Folland and A. Sitaram, “The uncertainty principle: a mathematical survey,” Journal of Fourier Analysis and Applications, vol. 3, no. 3, pp. 207–238, 1997.
2 V. Havin and B. J ¨oricke, The Uncertainty Principle in Harmonic Analysis, vol. 28 of Results in Mathematics and Related Areas (3), Springer, Berlin, Germany, 1994.
3 N. Bettaibi, A. Fitouhi, and W. Binous, “Uncertainty principles for the q-trigonometric Fourier transforms,” Mathematical Sciences Research Journal, vol. 11, no. 7, pp. 469–479, 2007.
4 G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 96 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, 2004.
5 V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
6 F. H. Jackson, “On aq-definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910.
7 M. E. H. Ismail, “On Jacksons thirdq-Bessel function andq-exponentials,” preprint, 2001.
8 T. H. Koornwinder and R. F. Swarttouw, “Onq-analogues of the Fourier and Hankel transforms,” Transactions of the American Mathematical Society, vol. 333, no. 1, pp. 445–461, 1992.
9 F. Bouzeffour, q-Analyse harmonique, Ph.D. thesis, Faculty of Sciences of Tunis, Tunis, Tunisia, 2002.
10 A. Fitouhi, N. Bettaibi, and W. Binous, “Inversion formulas for theq-Riemann-Liouville andq-Weyl transforms using wavelets,” Fractional Calculus & Applied Analysis, vol. 10, no. 4, pp. 327–342, 2007.
11 L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Upper Saddle River, NJ, USA, 2004.
12 W. G. Faris, “Inequalities and uncertainty principles,” Journal of Mathematical Physics, vol. 19, no. 2, pp. 461–466, 1978.
13 J. F. Price, “Inequalities and local uncertainty principles,” Journal of Mathematical Physics, vol. 24, no. 7, pp. 1711–1714, 1983.
14 J. F. Price, “Sharp local uncertainty inequalities,” Studia Mathematica, vol. 85, no. 1, pp. 37–45, 1986.
15 A. Fitouhi, N. Bettaibi, and K. Brahim, “The Mellin transform in quantum calculus,” Constructive Approximation, vol. 23, no. 3, pp. 305–323, 2006.
16 A. Fitouhi, M. M. Hamza, and F. Bouzeffour, “The q-jα Bessel function,” Journal of Approximation
