Volume 10 (2009), Issue 2, Article 42, 10 pp.
ON HEISENBERG AND LOCAL UNCERTAINTY PRINCIPLES FOR THE
q-DUNKL TRANSFORM
AHMED FITOUHI, FERJANI NOURI, AND SANA GUESMI
FACULTÉ DES S CIENCES DE T UNIS
1060 T UNIS , T UNISIA .
Ahmed.Fitouhi@fst.rnu.tn
I NSTITUT P RÉPARATOIRE AUX É TUDES D ’I NGÉNIEUR DE NABEUL
T UNISIA .
nouri.ferjani@yahoo.fr
FACULTÉ DES S CIENCES DE T UNIS
1060 T UNIS , T UNISIA .
guesmisana@yahoo.fr
Received 17 January, 2009; accepted 02 May, 2009
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we provide, for the q-Dunkl transform studied in [2], a Heisenberg
uncertainty principle and two local uncertainty principles leading to a new Heisenberg-Weyl type
inequality.
Key words and phrases: q-Dunkl transform, Heisenberg-Weyl inequality, Local uncertainty principles.
2000 Mathematics Subject Classification. 33D15, 26D10, 26D20, 39A13, 42A38.
1. I NTRODUCTION
In harmonic analysis, the uncertainty principle states that a function and its Fourier transform
cannot be simultaneously sharply localized. A quantitative formulation of this fact is provided
by the Heisenberg uncertainty principle, which asserts that every square integrable function f
on R verifies the following inequality
Z +∞
Z +∞
Z +∞
2
1
2
2
2 b
2
2
2
(1.1)
x |f (x)| dx
λ |f (λ)| dλ ≥
x |f (x)| dx ,
4
−∞
−∞
−∞
where
Z +∞
1
fb(λ) = √
f (x)e−iλx dx
2π −∞
is the classical Fourier transform.
Generalizations of this result in both classical and quantum analysis have been treated and
many versions of Heisenberg-Weyl type uncertainty inequalities were obtained for several generalized Fourier transforms (see [1], [14], [10]).
019-09
2
A. F ITOUHI , F. N OURI , AND S. G UESMI
In [2], by the use of the q 2 -analogue differential operator studied in [11], Bettaibi et al.
introduced a new q-analogue of the classical Dunkl operator and studied its related Fourier
transform, which is a q-analogue of the classical Bessel-Dunkl one and called the q-Dunkl
transform.
The aim of this paper is twofold: first, we prove a Heisenberg uncertainty principle for the
q-Dunkl transform and next, we state for this transform two local uncertainty principles leading
to a new q-Heisenberg-Weyl type inequality.
This paper is organized as follows: in Section 2, we present some preliminary notions and
notations useful in the sequel. In Section 3, we recall some results and properties from the
theory of the q-Dunkl operator and the q-Dunkl transform (see [2]). Section 4 is devoted to
proving a Heisenberg uncertainty principle for the q-Dunkl transform and as consequences, we
obtain Heisenberg uncertainty principles for the q 2 -analogue Fourier transform [12, 11] and
for the q-Bessel transform [2]. Finally, in Section 5, we state, for the q-Dunkl transform, two
local uncertainty principles, which give a new Heisenberg-Weyl type inequality for the q-Dunkl
transform.
2. N OTATIONS AND P RELIMINARIES
Throughout this paper, we assume q ∈]0, 1[, and refer to the general reference [6] for the definitions, notations and properties of the q-shifted factorials and the q-hypergeometric functions.
We write Rq = {±q n : n ∈ Z}, Rq,+ = {q n : n ∈ Z},
1 − qx
,
[x]q =
1−q
x∈C
and [n]q ! =
(q; q)n
,
(1 − q)n
n ∈ N.
The q 2 -analogue differential operator is (see [11, 12])

f q −1 z )+f (−q −1 z )−f (qz)+f (−qz)−2f (−z)

 (
if z =
6 0
2(1−q)z
(2.1)
∂q (f )(z) =

 lim ∂q (f )(x)
(in Rq )
if z = 0.
x→0
We remark that if f is differentiable at z, then limq→1 ∂q (f )(z) = f 0 (z).
A repeated application of the q 2 -analogue differential operator is denoted by:
∂q0 f = f,
∂qn+1 f = ∂q (∂qn f ).
The following lemma lists some useful computational properties of ∂q .
Lemma 2.1.
(1) For all functions f on Rq ,
∂q f (z) =
fe (q −1 z) − fe (z) fo (z) − fo (qz)
+
,
(1 − q)z
(1 − q)z
where, fe and fo are, respectively, the even and the odd parts of f .
(2) For two functions f and g on Rq , we have
• if f is even and g is odd,
∂q (f g)(z) = q∂q (f )(qz)g(z) + f (qz)∂q (g)(z)
= ∂q (g)(z)f (z) + qg(qz)∂q (f )(qz);
• if f and g are even,
∂q (f g)(z) = ∂q (f )(z)g(q −1 z) + f (z)∂q (g)(z).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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U NCERTAINTY P RINCIPLES FOR THE q-D UNKL T RANSFORM
3
The operator ∂q induces a q-analogue of the classical exponential function (see [11, 12])
e(z; q 2 ) =
(2.2)
∞
X
n=0
an
zn
,
[n]q !
a2n = a2n+1 = q n(n+1) .
with
The q-Jackson integrals are defined by (see [8])
Z a
∞
X
f (x)dq x = (1 − q)a
q n f (aq n ),
0
Z
n=0
b
Z
b
Z
0
0
Z
f (x)dq x,
0
f (x)dq x = (1 − q)
and
a
f (x)dq x −
f (x)dq x =
a
∞
Z
∞
X
q n f (q n ),
n=−∞
∞
f (x)dq x = (1 − q)
−∞
∞
X
n
n
q f (q ) + (1 − q)
n=−∞
∞
X
q n f (−q n ),
n=−∞
provided the sums converge absolutely.
The q-Gamma function is given by (see [8])
Γq (x) =
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
x 6= 0, −1, −2, ...
• Sq (Rq ) the space of functions f defined on Rq satisfying
∀n, m ∈ N,
Pn,m,q (f ) = sup xm ∂qn f (x) < +∞
x∈Rq
and
lim ∂qn f (x)
(in Rq )
exists;
n
o
• L∞
q (Rq ) = f : kf k∞,q = supx∈Rq |f (x)| < ∞ ;
R
p1
∞
p
p
2α+1
• Lα,q (Rq ) = f : kf kp,α,q = −∞ |f (x)| |x|
dq x < ∞ ;
R
p1
a
p
p
2α+1
• Lα,q ([−a, a]) = f : kf kp,α,q = −a |f (x)| |x|
dq x < ∞ .
x→0
For the particular case p = 2, we denote by h·; ·i the inner product of the Hilbert space
L2α,q (Rq ).
3. T HE q-D UNKL O PERATOR AND THE q-D UNKL T RANSFORM
In this section, we collect some basic properties of the q-Dunkl operator and the q-Dunkl
transform introduced in [2] which will useful in the sequel.
For α ≥ − 12 , the q-Dunkl operator is defined by
Λα,q (f )(x) = ∂q [Hα,q (f )] (x) + [2α + 1]q
f (x) − f (−x)
,
2x
where
Hα,q : f = fe + fo 7−→ fe + q 2α+1 fo .
It satisfies the following relations:
• For α = − 12 , Λα,q = ∂q .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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4
A. F ITOUHI , F. N OURI , AND S. G UESMI
• Λα,q lives Sq (Rq ) invariant.
• If f is odd then Λα,q (f )(x) = q 2α+1 ∂q f (x) + [2α + 1]q f (x)
and if f is even then
x
Λα,q (f )(x) = ∂q f (x).
• For all a ∈ C, Λα,q [f (ax)] = aΛα,q (f )(ax).
R +∞
• For all f and g such that −∞ Λα,q (f )(x)g(x)|x|2α+1 dq x exists, we have
Z +∞
Z +∞
2α+1
(3.1)
Λα,q (f )(x)g(x)|x|
dq x = −
Λα,q (g)(x)f (x)|x|2α+1 dq x.
−∞
−∞
It was shown in [2] that for each λ ∈ C, the function
ψλα,q : x 7−→ jα (λx; q 2 ) +
(3.2)
iλx
jα+1 (λx; q 2 )
[2α + 2]q
is the unique solution of the q-differential-difference equation:
(
Λα,q (f ) = iλf
f (0) = 1,
where jα (·; q 2 ) is the normalized third Jackson’s q-Bessel function given by
2
jα (x; q ) =
(3.3)
∞
X
(−1)n
n=0
The function
ψλα,q (x),
q n(n+1)
((1 − q)x)2n .
(q 2 ; q 2 )n (q 2(α+1) ; q 2 )n
has a unique extension to C × C and verifies the following properties.
α,q
α,q
• ψaλ
(x) = ψλα,q (ax) = ψax
(λ),
∀a, x, λ ∈ C.
• For all x, λ ∈ Rq ,
|ψλα,q (x)| ≤
(3.4)
4
.
(q; q)∞
The q-Dunkl transform FDα,q is defined on L1α,q (Rq ) (see [2]) by
Z
cα,q +∞
α,q
α,q
FD (f )(λ) =
f (x)ψ−λ
(x)|x|2α+1 dq x,
2 −∞
where
cα,q =
(1 + q)−α
.
Γq2 (α + 1)
It satisfies the following properties:
• For α = − 12 , FDα,q is the q 2 -analogue Fourier transform fb(·; q 2 ) given by (see [12, 11])
1/2 Z +∞
(1
+
q)
2
f (x)e(−iλx; q 2 )dq x.
fb(λ; q ) =
1
2
2Γq 2
−∞
• On the even functions space, FDα,q coincides with the q-Bessel transform given by (see
[2])
Z +∞
Fα,q (f )(λ) = cα,q
f (x)jα (λx; q 2 )x2α+1 dq x.
0
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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U NCERTAINTY P RINCIPLES FOR THE q-D UNKL T RANSFORM
5
• For all f ∈ L1α,q (Rq ), we have:
kFDα,q (f )k∞,q ≤
(3.5)
2cα,q
kf k1,α,q .
(q; q)∞
• For all f ∈ L1α,q (Rq ), such that xf ∈ L1α,q (Rq ),
FDα,q (Λα,q f )(λ) = iλFDα,q (f )(λ)
(3.6)
and
Λα,q (FDα,q (f )) = −iFDα,q (xf ).
(3.7)
• The q-Dunkl transform FDα,q is an isomorphism from L2α,q (Rq ) (resp. Sq (Rq )) onto itself
and satisfies the following Plancherel formula:
kFDα,q (f )k2,α,q = kf k2,α,q ,
(3.8)
f ∈ L2α,q (Rq ).
4. q-A NALOGUE OF THE H EISENBERG I NEQUALITY
In this section, we provide a Heisenberg uncertainty principle for the q-Dunkl transform.
For this purpose, inspired by the approach given in [10], we follow the steps of [1], using the
operator Λα,q instead of the operator ∂q , and consider the operators
Lα,q (f )(x) = fe (x) + q 2α+2 fo (qx) and
Qf (x) = xf (x),
and the q-commutator:
[Dα,q , Q]q = Dα,q Q − qQDα,q ,
where
Dα,q = Lα,q Λα,q .
The following theorem gives a Heisenberg uncertainty principle for the q-Dunkl transform
FDα,q .
Theorem 4.1. For f ∈ Sq (Rq ), we have
q 2α+1
[2α
+
1]
q
2
2
qkf k2,α,q + 1 − q −
(4.1)
kf
k
o
2,α,q
1 + q + q α−1 + q α
q 2α
≤ kxf k2,α,q kxFDα,q (f )(x)k2,α,q .
Proof. By Lemma 2.1 and simple calculus, we obtain
[2α + 1]q
2α+2
2α+1
[Dα,q , Q]q f = q
fe + q
1−
fo .
q 2α
Then, using the Cauchy-Schwarz inequality and the properties of the q-Dunkl operator, one can
write
2α+2
[2α
+
1]
q
2
2α+1
2
q
kf
k
+
q
1
−
kf
k
e
o
2,α,q
2,α,q
q 2α
= |h[Dα,q , Q]q f ; f i|
= |hDα,q Qf − qQDα,q f ; f i| ≤ |hDα,q Qf ; f i| + q |hQDα,q f ; f i|
= |hDα,q (xfe + xfo ); f i| + q |hDα,q f ; xf i|
= hΛα,q (xfe ) + q 2α+2 Λα,q (xfo )(qx); f i
+ q hq 2α+2 Λα,q (fe )(qx) + Λα,q (fo )(x); xf i
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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6
A. F ITOUHI , F. N OURI , AND S. G UESMI
≤ |hΛα,q (xfe ); f i| + q 2α+2 |hΛα,q (xfo )(qx); f i|
+ q 2α+3 |hΛα,q (fe )(qx); xf i| + q |hΛα,q (fo )(x); xf i|
= |hΛα,q (xfe ); f i| + q 2α+1 |hΛα,q (xfo (q.)); f i|
+ q 2α+2 |hΛα,q (fe (q.)); xf i| + q |hΛα,q (fo )(x); xf i|
≤ kxfe k2,α,q kxFDα,q (f )k2,α,q + q α−1 kxfo k2,α,q kxFDα,q (f )k2,α,q
+ q α kxfe k2,α,q kxFDα,q (f )k2,α,q + qkxfo k2,α,q kxFDα,q (f )k2,α,q
≤ (1 + q + q α−1 + q α )kxf k2,α,q kxFDα,q (f )k2,α,q ,
which achieves the proof.
As a consequence, we obtain a Heisenberg-Weyl uncertainty principle for the q 2 -analogue
Fourier transform (by taking α = −1/2) and the q-Bessel transform (in the even case).
Corollary 4.2.
(1) For f ∈ Sq (Rq ), we have
q
(4.2)
1 + q + q −3/2 + q −1/2
kf k22,q ≤ kxf k2,q kλfb(λ; q 2 )k2,q .
(2) For an even function f ∈ Sq (Rq ), we have
(4.3)
q 2α+2
kf k22,α,q ≤ kxf k2,α,q kλFα,q (f )(λ)k2,α,q .
1 + q + q α−1 + q α
We remark that when q tends to 1− , (4.2) tends at least formally to the classical Heisenberg
uncertainty principle given by (1.1).
5. L OCAL U NCERTAINTY P RINCIPLES
In this section, we will state, for the q-Dunkl transform, two local uncertainty principles
leading to a new Heisenberg-Weyl type inequality.
Notations: For E ⊂ Rq and f defined on Rq , we write
Z
Z ∞
Z
f (t)dq t =
f (t)χE (t)dq t and |E|α =
|t|2α+1 dq t,
−∞
E
E
where χE is the characteristic function of E.
Theorem 5.1. If 0 < a < α + 1, then for all bounded subsets E of Rq and all f ∈ L2α,q (Rq ),
we have
Z
a
(5.1)
|FDα,q (f )(λ)|2 |λ|2α+1 dq λ ≤ Ka,α |E| α+1 kxa f k22,α,q ,
E
where
Ka,α =
and e
cα,q =
2e
cα,q
p
[2(α + 1 − a)]q
α+1−a
a
2a
! α+1
α+1
α+1−a
2
2cα,q
.
(q;q)∞
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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U NCERTAINTY P RINCIPLES FOR THE q-D UNKL T RANSFORM
7
Proof. For r > 0, let χr = χ[−r,r] the characteristic function of [−r, r] and χ
er = 1 − χr .
1
Then for r > 0, we have, since f · χr ∈ Lq (Rq ),
1/2
Z
α,q
2 2
2α+1
|FD (f )(λ; q )| |λ|
dq λ
= kFDα,q (f ) · χE k2,α,q
E
≤ kFDα,q (f · χr )χE k2,α,q + kFDα,q (f · χ
er )χE k2,α,q
α,q
α,q
er )k2,α,q .
≤ |E|1/2
α kFD (f · χr )k∞,q + kFD (f · χ
Now, on the one hand, we have by the relation (3.5) and the Cauchy-Schwartz inequality,
cα,q kf · χr k1,α,q
kFDα,q (f.χr )k∞,q ≤ e
=e
cα,q kx−a χr · xa f k1,α,q
≤e
cα,q kx−a χr k2,α,q kxa f k2,α,q
2e
cα,q
≤p
r(α+1)−a kxa f k2,α,q .
[2(α + 1 − a)]q
On the other hand, since f ∈ L2α,q (Rq ), we have f · χ
er ∈ L2α,q (Rq ) and by the Plancherel
formula, we obtain
kFDα,q (f.e
χr )k2,α,q = kf · χ
er k2,α,q
= kx−a χ
er · xa f k2,α,q
≤ kx−a χ
er k∞,q kxa f k2,α,q
≤ r−a kxa f k2,α,q .
So,
Z
|FDα,q (f )(λ)|2 |λ|2α+1 dq λ
21
E
≤
1
2e
cα,q
p
|E|α2 rα+1−a + r−a
[2(α + 1 − a)]q
!
kxa f k2,α,q .
The desired result is obtained by minimizing the right hand side of the previous inequality over
r > 0.
Corollary 5.2. For α ≥ − 12 ,
0 < a < α + 1 and b > 0, we have for all f ∈ L2α,q (Rq ),
(a+b)
kf k2,α,q ≤ Ka,b,α kxa f kb2,α,q kλb FDα,q (f )ka2,α,q ,
(5.2)
with
Ka,b,α
" a
# a+b
2
b
b
b a+b a a+b
q −(2α+1)(a+b)
=
+
(2Ka,α ) 2
ab
a
b
([2α + 2]q ) 2(α+1)
where Ka,α is the constant given in Theorem 5.1.
er the supplementary of Er in Rq .
Proof. For r > 0, we put Er =] − r, r[∩Rq and E
r 2α+2
. Then the Plancherel formula and
We have Er is a bounded subset of Rq and |Er |α ≤ 2 [2α+2]
q
the previous theorem lead to
kf k22,α,q = kFDα,q (f )k22,α,q
Z
Z
α,q
2
2α+1
=
|FD (f )| (λ)|λ|
dq λ +
Er
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
|FDα,q (f )|2 (λ)|λ|2α+1 dq λ
er
E
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8
A. F ITOUHI , F. N OURI , AND S. G UESMI
a
≤ 2Ka,α |Er |αα+1 kxa f k22,α,q + r−2b kλb FDα,q (f )k22,α,q
Ka,α
2a
≤2
kxa f k22,α,q + r−2b kλb FDα,q (f )k22,α,q .
a r
α+1
[2α + 2]q
The desired result follows by minimizing the right expressions over r > 0.
0
Theorem 5.3. For α ≥ − 12 and a > α + 1, there exists a constant Ka,α,q
such that for all
2
bounded subsets E of Rq and all f in Lα,q (Rq ), we have
Z
2 α+1
2(1− α+1 )
0
a
(5.3)
|FDα,q (f )(λ)|2 |λ|2α+1 dq λ ≤ Ka,α,q
|E|α kf k2,α,q a kxa f k2,α,q
.
E
The proof of this result needs the following lemmas.
Lemma 5.4. Suppose a > α + 1, then for all f ∈ L2α,q (Rq ) such that xa f ∈ L2α,q (Rq ),
(5.4)
kf k21,α,q ≤ K2 kf k22,α,q + kxa f k22,α,q ,
where
(q 2a , q 2a , −q 2α+2 , −q 2(a−α−1) ; q 2a )∞
K2 = 2(1 − q) 2α+2 2(a−α−1)
.
(q
,q
, −q 2a , −1; q 2a )∞
Proof. From ([4, Example 1]) and Hölder’s inequality, we have
Z +∞
2
2
2a 12
2a − 12
2α+1
kf k1,α,q =
(1 + |x| ) |f (x)|(1 + |x| ) |x|
dq x
−∞
≤ K2 kf k22,α,q + kxa f k22,α,q ,
where
+∞
x2α+1
dq x
1 + x2a
0
(q 2a , q 2a , −q 2α+2 , −q 2(a−α−1) ; q 2a )∞
= 2(1 − q) 2α+2 2(a−α−1)
.
(q
,q
, −q 2a , −1; q 2a )∞
Z
K2 = 2
Lemma 5.5. Suppose a > α + 1, then for all f ∈ L2α,q (Rq ) such that xa f ∈ L2α,q (Rq ), we have
(1− α+1 )
kf k1,α,q ≤ K3 kf k2,α,q a
(5.5)
α+1
a
kxa f k2,α,q
,
where
"
K3 = K3 (a, α, q) =
# 12
α+1
a
a
α
+
1
q 2(α+1) (
− 1)
q −2(α+1) (1 +
) K2 .
α+1
a−α−1
Proof. For s ∈ Rq , define the function fs by fs (x) = f (sx), x ∈ Rq .
We have
kfs k1,α,q = s−2(α+1) kf k1,α,q ,
kxa fs k22,α,q = s−2(α+a+1) kxa f k22,α,q .
Replacement of f by fs in Lemma 5.4 gives:
kf k21,α,q ≤ K2 s2(α+1) kf k22,α,q + s2(α−a+1) kxa f k22,α,q .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
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U NCERTAINTY P RINCIPLES FOR THE q-D UNKL T RANSFORM
Now, for all r > 0, put α(r) =
Then, for all r > 0,
kf k21,α,q
Log(r)
Log(q)
−E
Log(r)
Log(q)
. We have s =
9
r
q α(r)
∈ Rq and r ≤ s < qr .
" #
2(α+1)
r
≤ K2
kf k22,α,q + r2(α−a+1) kxa f k22,α,q .
q
The right hand side of this inequality is minimized by choosing
2a1
1
α+1
a
− a1
a
kxa f k2,α,q
.
r=
−1
q a kf k2,α,q
α+1
When this is done we obtain the result.
Proof of Theorem 5.3. Let E be a bounded subset of Rq . When the right hand side of the inequality is finite, Lemma 5.4 implies that f ∈ L1q (Rq ), so, FDα,q (f ) is defined and bounded on
Rq . Using Lemma 5.5, the relation (3.5) and the fact that
Z
|FDα,q (f )(λ)|2 |λ|2α+1 dq λ ≤| E |α kFDα,q (f )k2∞,q ,
E
we obtain the result with
4(1 + q)−2α
0
= 2
K2
Ka,α,q
Γq2 (α + 1)(q; q)2∞ 3
α+1
a
8(1 − q)(1 + q)−2α
a
α+1
2(α+1)
−2(α+1)
= 2
q
−1
q
1+
Γq2 (α + 1)(q; q)2∞
α+1
a−α−1
×
(q 2a , q 2a , −q 2α+2 , −q 2(a−α−1) ; q 2a )∞
.
(q 2α+2 , q 2(a−α−1) , −q 2a , −1; q 2a )∞
Corollary 5.6. For α ≥ − 12 , a > α + 1 and b > 0, we have for all f ∈ L2α,q (Rq ),
(a+b)
0
kf k2,α,q ≤ Ka,b,α
kxa f kb2,α,q kλb FDα,q (f )ka2,α,q ,
(5.6)
with
0
Ka,b,α
=
0
Ka,α,q
[2α + 2]q
ab
2α+2
q −(4α+2)
"
b
α+1
α+1
α+b+1
+
b
α+1
#! a(α+b+1)
b
− α+b+1
2(α+1)
,
0
where Ka,α,q
is the constant given in the previous theorem.
Proof. The same techniques as in Corollary 5.2 give the result.
The following result gives a new Heisenberg-Weyl type inequality for the q-Dunkl transform.
Theorem 5.7. For α ≥ − 12 , α 6= 0, we have for all f ∈ L2α,q (Rq ),
(5.7)
kf k22,α,q ≤ Kα kxf k2,α,q kλFDα,q (f )k2,α,q ,
with
(
Kα =
K1,1,α if α > 0
0
K1,1,α
if α < 0.
Proof. The result follows from Corollaries 5.2 and 5.6, by taking a = b = 1.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
http://jipam.vu.edu.au/
10
A. F ITOUHI , F. N OURI ,
AND
S. G UESMI
Remark 1. Note that Theorem 4.1 and Theorem 5.7 are both Heisenberg-Weyl type inequalities
for the q-Dunkl transform. However, the constants in the two theorems are different, the first
one seems to be more optimal. Moreover, Theorem 4.1 is true for every α > − 12 and uses both
f and f0 , in contrast to Theorem 5.7, which is true only for α 6= 0 and uses only f .
R EFERENCES
[1] N. BETTAIBI, Uncertainty principles in q 2 -analogue Fourier analysis, Math. Sci. Res. J., 11(11)
(2007), 590–602.
[2] N. BETTAIBI AND R.H. BETTAIEB, q-Aanalogue of the Dunkl transform on the real line, to
appear in Tamsui Oxford J. Mathematical Sciences.
[3] M.G. COWLING AND J.F. PRICE, Generalisation of Heisenberg inequality, Lecture Notes in
Math., 992, Springer, Berlin, (1983), 443-449.
[4] A. FITOUHI, N. BETTAIBI AND K. BRAHIM, Mellin transform in quantum calculus, Constructive Approximation, 23(3) (2006), 305–323.
[5] G.B. FOLLAND AND A. SITARAM, The uncertainty principle: A mathematical survey, J. Fourier
Anal. and Applics., 3(3) (1997), 207–238.
[6] G. GASPER AND M. RAHMAN, Basic hypergeometric series, Encyclopedia of Mathematics and
its Applications, 35 (1990), Cambridge Univ. Press, Cambridge, UK.
[7] V. HAVIN AND B. JÖRICKE, Uncertainty Principle in Harmonic Analysis, Springer-Verlag,
Berlin, (1994).
[8] F.H. JACKSON, On a q-definite integrals, Quart. J. Pure and Appl. Math., 41 (1910), 193–203.
[9] V.G. KAC AND P. CHEUNG, Quantum Calculus, Universitext, Springer-Verlag, New York, (2002).
[10] M. RÖSLER AND M. VOIT, Uncertainty principle for Hankel transforms, Proc. Amer. Math. Soc.,
127 (1999), 183–194.
[11] R.L. RUBIN, A q 2 -analogue operator for q 2 -analogue Fourier analysis, J. Math. Analys. Appl., 212
(1997), 571–582.
[12] R.L. RUBIN, Duhamel solutions of non-homogenous q 2 -analogue wave equations, Proc. Amer.
Math. Soc., 13(3) (2007), 777–785.
[13] A. SITARAM, M. SUNDARI AND S. THANGAVALU, Uncertainty principle on certain Lie
groups, Proc. Indian Acad. Sci. (Math. Sci.), 105 (1995), 135–151.
[14] R.S. STRICHARTZ, Uncertainty principle in harmonic analysis, J. Functional Analysis, 84 (1989),
97–114.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 42, 10 pp.
http://jipam.vu.edu.au/