HEISENBERG UNCERTAINTY PRINCIPLES FOR
SOME q 2-ANALOGUE FOURIER TRANSFORMS
q-Heisenberg Uncertainty
Principles
WAFA BINOUS
Institut De Bio-technologie de Béjà
Béjà, Tunisia.
EMail: wafabinous@yahoo.fr
Wafa Binous
vol. 9, iss. 2, art. 47, 2008
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Contents
Received:
07 December, 2007
Accepted:
20 May, 2008
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Communicated by:
S.S. Dragomir
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2000 AMS Sub. Class.:
33D15, 26D10, 26D15.
Key words:
Heisenberg inequality, q-Fourier transforms.
Abstract:
The aim of this paper is to state q-analogues of the Heisenberg uncertainty principles for some q 2 -analogue Fourier transforms introduced and studied in [7, 8].
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Contents
1
Introduction
3
2
Notations and Preliminaries
4
3
q-Analogue of the Heisenberg Uncertainly Principle
9
q-Heisenberg Uncertainty
Principles
Wafa Binous
vol. 9, iss. 2, art. 47, 2008
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1.
Introduction
One of the most famous uncertainty principles is the so-called Heisenberg uncertainty principle. With the use of an inequality involving a function and its Fourier
transform, it states that in classical Fourier analysis it is impossible to find a function
f that is arbitrarily well localized together with its Fourier transform fb.
In this paper, we will prove that similar to the classical theory, a non-zero function
and its q 2 -analogue Fourier transform (see [7, 8]) cannot both be sharply localized.
For this purpose we will prove a q-analogue of the Heisenberg uncertainly principle.
This paper is organized as follows: in Section 2, some notations, results and definitions from the theory of the q 2 -analogue Fourier transform are presented. All of
these results can be found in [7] and [8]. In Section 3, q-analogues of the Heisenberg
uncertainly principle are stated.
q-Heisenberg Uncertainty
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Wafa Binous
vol. 9, iss. 2, art. 47, 2008
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2.
Notations and Preliminaries
Throughout this paper, we will follow the notations of [7, 8]. We fix q ∈]0, 1[ such
that Log(1−q)
∈ 2Z. For the definitions, notations and properties of the q-shifted
Log(q)
factorials and the q-hypergeometric functions, refer to the book by G. Gasper and
M. Rahman [3].
Define
Rq = {±q n : n ∈ Z} and Rq,+ = {q n : n ∈ Z}.
We also denote
q-Heisenberg Uncertainty
Principles
Wafa Binous
vol. 9, iss. 2, art. 47, 2008
x
[x]q =
(2.1)
1−q
,
1−q
x∈C
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and
Contents
[n]q ! =
(2.2)
(q; q)n
,
(1 − q)n
n ∈ N.
The q 2 -analogue differential operator (see [8]) is
−1
(2.3)
∂q (f )(z) =
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−1
f (q z) + f (−q z) − f (qz) + f (−qz) − 2f (−z)
.
2(1 − q)z
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We remark that if f is differentiable at z, then limq→1 ∂q (f )(z) = f 0 (z).
∂q is closely related to the classical q-derivative operators studied in [3, 5].
The q-trigonometric functions q-cosine and q-sine are defined by (see [7, 8]):
(2.4)
cos(x; q 2 ) =
∞
X
n=0
(−1)n q n(n+1)
x2n
[2n]q !
Close
and
2
sin(x; q ) =
(2.5)
∞
X
(−1)n q n(n+1)
n=0
x2n+1
.
[2n + 1]q !
These functions induce a ∂q -adapted q 2 -analogue exponential function by
e(z; q 2 ) = cos(−iz; q 2 ) + i sin(−iz; q 2 ).
(2.6)
e(z; q 2 ) is absolutely convergent for all z in the plane since both of its component
functions are absolutely convergent. limq→1− e(z; q 2 ) = ez (exponential function)
pointwise and uniformly on compacta.
The q-Jackson integrals are defined by (see [4])
Z ∞
∞
X
(2.7)
f (x)dq x = (1 − q)
{f (q n ) + f (−q n )} q n
−∞
n=−∞
and
Z
∞
f (x)dq x = (1 − q)
(2.8)
0
∞
X
n
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q f (q ),
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n=−∞
−∞
(
Lpq (Rq,+ ) =
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n
provided that the sums converge absolutely. Using these q-integrals, we define for
p > 0,
)
(
Z ∞
p1
(2.9)
Lpq (Rq ) = f : kf kp,q =
|f (x)|p dq x
<∞ ,
(2.10)
q-Heisenberg Uncertainty
Principles
Z
f:
0
∞
|f (x)|p dq x
p1
)
<∞
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and
(
(2.11)
L∞
q (Rq )
)
f : kf k∞,q = sup |f (x)| < ∞ .
=
x∈Rq
The following result can be verified by direct computation.
R∞
Lemma 2.1. If −∞ f (t)dq t exists, then
R∞
R∞
1. for all integers n, −∞ f (q n t)dq t = q −n −∞ f (t)dq t;
R∞
2. f odd implies that −∞ f (t)dq t = 0;
R∞
R∞
3. f even implies that −∞ f (t)dq t = 2 0 f (t)dq t.
q-Heisenberg Uncertainty
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Wafa Binous
vol. 9, iss. 2, art. 47, 2008
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The following lemma lists some useful computational properties of ∂q , and reflects the sensitivity of this operator to the parity of its argument. The proof is
straightforward.
Lemma 2.2.
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1. If f is odd ∂q f (z) =
f (z)−f (qz)
(1−q)z
and if f is even ∂q f (z) =
f (q −1 z)−f (z)
.
(1−q)z
2. We have ∂q sin(x; q 2 ) = cos(x; q 2 ), ∂q cos(x; q 2 ) = − sin(x; q 2 ) and ∂q e(x; q 2 ) =
e(x; q 2 ).
3. If f and g are both odd, then
z
z
z
−1
∂q (f g)(z) = q (∂q f )
g(z) + q f
(∂q g)
.
q
q
q
−1
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4. If f is odd and g is even, then
∂q (f g)(z) = (∂q f ) (z) g(z) + qf (qz) (∂q g) (qz) .
5. If f and g are both even, then
z
∂q (f g)(z) = (∂q f )(z)g
+ f (z) (∂q g) (z) .
q
The following simple result, giving a q-analogue of the integration by parts theorem, can be verified by direct calculation.
R∞
Lemma 2.3. If −∞ (∂q f )(x)g(x)dq x exists, then
Z ∞
Z ∞
(2.12)
(∂q f )(x)g(x)dq x = −
f (x)(∂q g)(x)dq x.
−∞
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vol. 9, iss. 2, art. 47, 2008
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−∞
With the use of the q-Gamma function
(q; q)∞
Γq (x) = x
(1 − q)1−x ,
(q ; q)∞
R.L. Rubin defined in [8] the q 2 -analogue Fourier transform as
Z ∞
2
b
(2.13)
f (x; q ) = K
f (t)e(−itx; q 2 )dq t,
−∞
1
where K =
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(1+q) 2
.
2Γq2 ( 12 )
2
We define the q -analogue Fourier-cosine and Fourier-sine transform as (see [2]
and [6])
Z ∞
(2.14)
Fq (f )(x) = 2K
f (t) cos(xt; q 2 )dq t
0
and
∞
Z
(2.15)
q F(f )(x)
f (t) sin(xt; q 2 )dq t.
= 2K
0
Observe that if f is even then fb(·; q 2 ) = Fq and if f is odd then fb(·; q 2 ) =q F.
It was shown in [8] that we have the following theorem.
q-Heisenberg Uncertainty
Principles
Theorem 2.4.
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1. If f (u), uf (u) ∈ L1q (Rq ), then ∂q
fb (x; q 2 ) = (−iuf (u))b(x; q 2 ).
vol. 9, iss. 2, art. 47, 2008
2. If f, ∂q f ∈ L1q (Rq ), then (∂q f ) b(x; q 2 ) = ixfb (x; q 2 )
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3. For f ∈ L2q (Rq ), kfb (.; q 2 )k2,q = kf k2,q .
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3.
q-Analogue of the Heisenberg Uncertainly Principle
For a function f defined on Rq , we denote by f0 and fe its odd and even parts
respectively. Let us begin with the following theorem.
Theorem 3.1. If f , xf and xfb(x; q 2 ) are in L2q (Rq ), then
h i
3
3
(3.1)
kf k22,q ≤ kxfb(x; q 2 )k2,q q 1 + q − 2 kxfo k2,q + 1 + q 2 kxfe k2,q .
Proof. Using the properties of the q 2 -analogue differential operator ∂q , the properties
of the q-integrals, the Hölder inequality and Theorem 2.4, we can see that
Z ∞
x∂
(f
f
)(x)d
x
q
q
−∞
Z ∞
= x qf 0 (x) + f e (q −1 x) (∂q f )(x)dq x
−∞
Z ∞
+
x (qf0 (qx) + fe (x)) (∂q f )(x)dq x
−∞
Z ∞
Z ∞
≤q
|xf0 (x)||∂q f (x)|dq x +
|xfe (q −1 x)||∂q f (x)|dq x
−∞
−∞
Z ∞
Z ∞
+
|xfe (x)||∂q f (x)|dq x + q
|xf0 (x)||∂q f (x)|dq x
−∞
−∞
" Z
1 Z
1
∞
|xfo (x)|2 dq x
≤ k∂q f k2,q q
∞
2
|xfe (q −1 x)|2 dq x
+
−∞
Z
∞
+
−∞
|xfe (x)|2 dq x
−∞
12
Z
∞
+q
−∞
|xfo (qx)|2 dq x
12 #
2
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i
h 3
− 32
b
2
= kxf k2,q q 1 + q
kxfo k2,q + 1 + q kxfe k2,q .
On the other hand, using the q-integration by parts theorem, we obtain
Z ∞
Z ∞
x∂q (f f )(x)dq x = −
|f (x)|2 dq x = −kf k22,q ,
−∞
−∞
q-Heisenberg Uncertainty
Principles
which completes the proof.
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Corollary 3.2. If f , xf and xfb are in L2q (Rq ), then
(3.2)
kxf k2,q kxfb(x; q 2 )k2,q ≥
vol. 9, iss. 2, art. 47, 2008
1
q
− 12
+1+q+q
3
2
kf k22,q .
Proof. The properties of the q-integral imply
Z ∞
2
kxf k2,q =
x2 (fo (x) + fe (x)) f o (x) + f e (x) dq x
Z−∞
Z ∞
∞
2
=
x fo (x)f o (x)dq x +
x2 fe (x)f e (x)dq x
−∞
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−∞
= kxfo k22,q + kxfe k22,q .
So, kxfo k2,q ≤ kxf k2,q and kxfe k2,q ≤ kxf k2,q .
These inequalities together with the previous theorem give the desired result.
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Corollary 3.3.
1. If f , xf and xFq are in L2q (Rq,+ ), then
Z
∞
2
2
12 Z
0
2
2
x |f (x)| dq x
(3.3)
12
∞
x |Fq (x)| dq x
0
≥
Z
1
1+q
3
2
∞
|f (x)|2 dq x.
0
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2. If f , xf and x q F are in L2q (Rq,+ ), then
Z
∞
2
2
12 Z
x |f (x)| dq x
(3.4)
0
q-Heisenberg Uncertainty
Principles
vol. 9, iss. 2, art. 47, 2008
∞
x2 | q F(x)|2 dq x
12
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0
≥
Z
1
q 1+q
− 32
∞
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|f (x)|2 dq x.
0
Proof. The proof is a simple application of the previous theorem on taking g(x) =
f (x) if x is positive and g(x) = f (−x) (resp. g(x) = −f (−x)) if not in the first
case (resp. second case).
Remark 1. Corollary 3.2 gives a q-analogue of the Heisenberg uncertainty principle
for the q 2 -analogue Fourier transform fb(·; q 2 ).
Remark 2. Corollary 3.3 gives a q-analogue of the Heisenberg uncertainty principles
for the q 2 -analogue Fourier-cosine and Fourier-sine transforms. These inequalities
are slightly different from those given in [1]. This is due to the related q-analogue of
special functions used.
Remark 3. Note that when q tends to 1, these inequalities tend at least formally to
the corresponding classical ones.
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References
[1] N. BETTAIBI, A. FITOUHI AND W. BINOUS, Uncertainty principle for the
q-trigonometric Fourier transforms, Math. Sci. Res. J., 11(7) (2007), 469–479.
[2] F. BOUZEFFOUR, q-Cosine Fourier Transform and q-Heat Equation, Ramanujan Journal.
[3] G. GASPER AND M. RAHMAN, Basic Hypergeometric Series, Encyclopedia of
Mathematics and its Applications, Vol. 35, Cambridge Univ. Press, Cambridge,
UK, 1990.
[4] F.H. JACKSON, On a q-definite integrals, Quarterly Journal of Pure and Applied
Mathematics, 41 (1910), 193-203.
[5] V.G. KAC AND P. CHEUNG, Quantum Calculus, Universitext, Springer-Verlag,
New York, (2002).
[6] T.H. KOORNWINDER AND R.F. SWARTTOUW, On q-analogues of the
Fourier and Hankel transforms, Trans. Amer. Math. Soc., 333 (1992), 445–461.
2
q-Heisenberg Uncertainty
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vol. 9, iss. 2, art. 47, 2008
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2
[7] R.L. RUBIN, A q -Analogue Operator for q -analogue Fourier Analysis, J.
Math. Analys. App., 212 (1997), 571–582.
[8] R.L. RUBIN, Duhamel Solutions of non-Homogenous q 2 -Analogue Wave Equations, Proc. of Amer. Math. Soc., 135(3) (2007), 777–785.
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