HEISENBERG UNCERTAINTY PRINCIPLES FOR SOME q2-ANALOGUE FOURIER TRANSFORMS

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q-Heisenberg Uncertainty Principles Wafa Binous vol. 9, iss. 2, art. 47, 2008

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HEISENBERG UNCERTAINTY PRINCIPLES FOR SOME q

²

-ANALOGUE FOURIER TRANSFORMS

WAFA BINOUS

Institut De Bio-technologie de Béjà Béjà, Tunisia.

EMail:wafabinous@yahoo.fr

Received: 07 December, 2007

Accepted: 20 May, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 33D15, 26D10, 26D15.

Key words: Heisenberg inequality,q-Fourier transforms.

Abstract: The aim of this paper is to stateq-analogues of the Heisenberg uncertainty principles for someq²-analogue Fourier transforms introduced and studied in [7,8].

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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 transformf.b

In this paper, we will prove that similar to the classical theory, a non-zero function and itsq²-analogue Fourier transform (see [7,8]) cannot both be sharply localized.

For this purpose we will prove aq-analogue of the Heisenberg uncertainly principle.

This paper is organized as follows: in Section 2, some notations, results and definitions from the theory of theq²-analogue Fourier transform are presented. All of these results can be found in [7] and [8]. In Section3,q-analogues of the Heisenberg uncertainly principle are stated.

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2. Notations and Preliminaries

Throughout this paper, we will follow the notations of [7, 8]. We fixq ∈]0,1[such that ^Log(1−q)_Log(q) ∈ 2Z. For the definitions, notations and properties of the q-shifted factorials and the q-hypergeometric functions, refer to the book by G. Gasper and M. Rahman [3].

Define

Rq ={±qⁿ:n∈Z} and Rq,+={qⁿ :n ∈Z}.

We also denote

(2.1) [x]_q = 1−q^x

1−q , x∈C and

(2.2) [n]_q! = (q;q)_n

(1−q)ⁿ, n ∈N. Theq²-analogue differential operator (see [8]) is

(2.3) ∂_q(f)(z) = f(q⁻¹z) +f(−q⁻¹z)−f(qz) +f(−qz)−2f(−z)

2(1−q)z .

We remark that iff is differentiable atz, thenlimq→1∂q(f)(z) = f⁰(z).

∂q is closely related to the classicalq-derivative operators studied in [3,5].

Theq-trigonometric functionsq-cosine andq-sine are defined by (see [7,8]):

(2.4) cos(x;q²) =

∞

X

n=0

(−1)ⁿqⁿ⁽ⁿ⁺¹⁾ x²ⁿ [2n]_q!

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and

(2.5) sin(x;q²) =

∞

X

n=0

(−1)ⁿqⁿ⁽ⁿ⁺¹⁾ x²ⁿ⁺¹ [2n+ 1]_q!.

These functions induce a∂_q-adaptedq²-analogue exponential function by (2.6) e(z;q²) = cos(−iz;q²) +isin(−iz;q²).

e(z;q²) is absolutely convergent for allz in the plane since both of its component functions are absolutely convergent. lim_q→1⁻e(z;q²) = e^z (exponential function) pointwise and uniformly on compacta.

Theq-Jackson integrals are defined by (see [4]) (2.7)

Z ∞

−∞

f(x)d_qx= (1−q)

∞

X

n=−∞

{f(qⁿ) +f(−qⁿ)}qⁿ

and (2.8)

Z ∞

0

f(x)d_qx= (1−q)

∞

X

n=−∞

qⁿf(qⁿ),

provided that the sums converge absolutely. Using these q-integrals, we define for p >0,

(2.9) L^p_q(Rq) = (

f :kfk_p,q = Z ∞

−∞

|f(x)|^pd_qx _p¹

<∞ )

,

(2.10) L^p_q(Rq,+) = (

f : Z ∞

0

|f(x)|^pd_qx ¹_p

<∞ )

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and

(2.11) L^∞_q (R^q) = (

f :kfk_∞,q = sup

x∈Rq

|f(x)|<∞ )

.

The following result can be verified by direct computation.

Lemma 2.1. IfR∞

−∞f(t)d_qtexists, then 1. for all integersn,R∞

−∞f(qⁿt)d_qt=q⁻ⁿR∞

−∞f(t)d_qt;

2. f odd implies thatR∞

−∞f(t)d_qt= 0;

3. f even implies thatR∞

−∞f(t)dqt = 2R∞

0 f(t)dqt.

The following lemma lists some useful computational properties of ∂_q, and re- flects the sensitivity of this operator to the parity of its argument. The proof is straightforward.

Lemma 2.2.

1. Iff is odd∂_qf(z) = ^f^(z)−f(qz)_(1−q)z and iff is even∂_qf(z) = ^f(q⁻¹_(1−q)z^z)−f(z).

2. We have∂_qsin(x;q²) = cos(x;q²),∂_qcos(x;q²) =−sin(x;q²)and∂_qe(x;q²) = e(x;q²).

3. Iff andg are both odd, then

∂_q(f g)(z) =q⁻¹(∂_qf) z

q

g(z) +q⁻¹f z

q

(∂_qg) z

q

.

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4. Iff is odd andgis even, then

∂_q(f g)(z) = (∂_qf) (z)g(z) +qf(qz) (∂_qg) (qz). 5. Iff andg are both even, then

∂_q(f g)(z) = (∂_qf)(z)g z

q

+f(z) (∂_qg) (z).

The following simple result, giving aq-analogue of the integration by parts theorem, can be verified by direct calculation.

Lemma 2.3. IfR∞

−∞(∂_qf)(x)g(x)d_qxexists, then (2.12)

Z ∞

−∞

(∂_qf)(x)g(x)d_qx=− Z ∞

−∞

f(x)(∂_qg)(x)d_qx.

With the use of theq-Gamma function Γ_q(x) = (q;q)∞

(q^x;q)∞

(1−q)^1−x,

R.L. Rubin defined in [8] theq²-analogue Fourier transform as (2.13) fb(x;q²) = K

Z ∞

−∞

f(t)e(−itx;q²)d_qt,

whereK = ^(1+q)

12

2Γ_q2(¹2).

We define theq²-analogue Fourier-cosine and Fourier-sine transform as (see [2]

and [6])

(2.14) F_q(f)(x) = 2K

Z ∞

0

f(t) cos(xt;q²)d_qt

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and

(2.15) _qF(f)(x) = 2K

Z ∞

0

f(t) sin(xt;q²)d_qt.

Observe that iff is even thenfb(·;q²) = F_qand iff is odd thenfb(·;q²) =_q F. It was shown in [8] that we have the following theorem.

Theorem 2.4.

1. Iff(u), uf(u)∈L¹_q(Rq), then∂_q fb

(x;q²) = (−iuf(u))b(x;q²).

2. Iff, ∂_qf ∈L¹_q(Rq), then(∂_qf)b(x;q²) =ixfb(x;q²) 3. Forf ∈L²_q(Rq),kfb(.;q²)k_2,q =kfk_2,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 f_e its odd and even parts respectively. Let us begin with the following theorem.

Theorem 3.1. Iff,xf andxfb(x;q²)are inL²_q(R^q), then (3.1) kfk²_2,q ≤ kxfb(x;q²)k2,q

h q

1 +q⁻³²

kxfok2,q +

1 +q³²

kxfek2,q

i .

Proof. Using the properties of theq²-analogue differential operator∂_q, the properties of theq-integrals, the Hölder inequality and Theorem2.4, we can see that

Z ∞

−∞

x∂_q(f f)(x)d_qx

=

Z ∞

−∞

x qf₀(x) +f_e(q⁻¹x)

(∂_qf)(x)d_qx

+ Z ∞

−∞

x(qf0(qx) +fe(x)) (∂qf)(x)dqx

≤q Z ∞

−∞

|xf₀(x)||∂_qf(x)|d_qx+ Z ∞

−∞

|xf_e(q⁻¹x)||∂_qf(x)|d_qx

+ Z ∞

−∞

|xf_e(x)||∂_qf(x)|d_qx+q Z ∞

−∞

|xf₀(x)||∂_qf(x)|d_qx

≤ k∂_qfk_2,q

"

q Z ∞

−∞

|xf_o(x)|²d_qx ¹₂

+ Z ∞

−∞

|xf_e(q⁻¹x)|²d_qx ¹₂

+ Z ∞

−∞

|xf_e(x)|²d_qx ¹₂

+q Z ∞

−∞

|xf_o(qx)|²d_qx ¹₂#

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=kxfkb _2,qh q

1 +q⁻³²

kxf_ok_2,q+

1 +q³²

kxf_ek_2,qi .

On the other hand, using theq-integration by parts theorem, we obtain Z ∞

−∞

x∂_q(f f)(x)d_qx=− Z ∞

−∞

|f(x)|²d_qx=−kfk²_2,q,

which completes the proof.

Corollary 3.2. Iff,xf andxfbare inL²_q(R^q), then

(3.2) kxfk_2,qkxf(x;b q²)k_2,q ≥ 1

q⁻¹² + 1 +q+q³²kfk²_2,q. Proof. The properties of theq-integral imply

kxfk²_2,q = Z ∞

−∞

x²(f_o(x) +f_e(x)) f_o(x) +f_e(x) d_qx

= Z ∞

−∞

x²f_o(x)f_o(x)d_qx+ Z ∞

−∞

x²f_e(x)f_e(x)d_qx

=kxf_ok²_2,q +kxf_ek²_2,q. So,kxfok2,q ≤ kxfk2,qandkxfek2,q ≤ kxfk2,q.

These inequalities together with the previous theorem give the desired result.

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Corollary 3.3.

1. Iff,xf andxF_q are inL²_q(Rq,+), then

(3.3)

Z ∞

0

x²|f(x)|²d_qx

¹₂ Z ∞

0

x²|F_q(x)|²d_qx ¹₂

≥ 1

1 +q³² Z ∞

0

|f(x)|²d_qx.

2. Iff,xf andx_qF are inL²_q(Rq,+), then

(3.4)

Z ∞

0

x²|f(x)|²d_qx

¹₂ Z ∞

0

x²|_qF(x)|²d_qx ¹₂

≥ 1

q

1 +q⁻³² Z ∞

0

|f(x)|²d_qx.

Proof. The proof is a simple application of the previous theorem on takingg(x) = f(x) ifx is positive and g(x) = f(−x)(resp. g(x) = −f(−x)) if not in the first case (resp. second case).

Remark 1. Corollary3.2gives aq-analogue of the Heisenberg uncertainty principle for theq²-analogue Fourier transformf(·;b q²).

Remark 2. Corollary3.3gives aq-analogue of the Heisenberg uncertainty principles for the q²-analogue Fourier-cosine and Fourier-sine transforms. These inequalities are slightly different from those given in [1]. This is due to the relatedq-analogue of special functions used.

Remark 3. Note that whenq 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 andq-Heat Equation, Ramanu- jan Journal.

[3] G. GASPERANDM. RAHMAN, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge Univ. Press, Cambridge, UK, 1990.

[4] F.H. JACKSON, On aq-definite integrals, Quarterly Journal of Pure and Applied Mathematics, 41 (1910), 193-203.

[5] V.G. KACANDP. 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.

[7] R.L. RUBIN, A q²-Analogue Operator for q²-analogue Fourier Analysis, J.

Math. Analys. App., 212 (1997), 571–582.

[8] R.L. RUBIN, Duhamel Solutions of non-Homogenousq²-Analogue Wave Equa- tions, Proc. of Amer. Math. Soc., 135(3) (2007), 777–785.