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volume 7, issue 5, article 171, 2006.

Received 17 April, 2006;

accepted 28 July, 2006.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

INEQUALITIES IN q−FOURIER ANALYSIS

LAZHAR DHAOUADI, AHMED FITOUHI AND J. EL KAMEL

Ecole Préparatoire d’Ingénieur Bizerte, Tunisia.

EMail:lazhardhaouadi@yahoo.fr Faculté des Sciences de Tunis 1060 Tunis, Tunisia.

EMail:Ahmed.Fitouhi@fst.rnu.tn

c

2000Victoria University ISSN (electronic): 1443-5756 115-06

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Inequalities inq−Fourier Analysis

Lazhar Dhaouadi, Ahmed Fitouhi and J. El Kamel

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J. Ineq. Pure and Appl. Math. 7(5) Art. 171, 2006

http://jipam.vu.edu.au

Abstract

In this paper we introduce theq−Bessel Fourier transform, theq−Bessel translation operator and theq−convolution product. We prove that theq−heat semigroup is contractive and we establish theq−analogue of Babenko inequalities associated to theq−Bessel Fourier transform. With applications and finally we enunciate aq−Bessel version of the central limit theorem.

2000 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary 26D10.

Key words: ˇCebyšev functional, Grüss inequality, Bessel, Beta and Zeta function bounds.

1. Introduction and Preliminaries

In introducingq−Bessel Fourier transforms, the q−Bessel translation operator and theq−convolution product we shall use the standard conventional notation as described in [4]. For further detailed information onq−derivatives, Jackson q−integrals and basic hypergeometric series we refer the interested reader to [4], [10], and [8].

The following two propositions will useful for the remainder of the paper.

Proposition 1.1. Consider0< q <1. The series (w;q)∞1φ₁(0, w;q;z) =

∞

X

n=0

(−1)ⁿqⁿ⁽ⁿ⁻¹⁾² (wqⁿ;q)∞

(q;q)_n zⁿ,

defines an entire analytic function inz, w, which is also symmetric inz, w:

(w;q)_∞1φ₁(0, w;q;z) = (z;q)_∞1φ₁(0, z;q;w).

Both sides can be majorized by

|(w;q)∞1φ₁(0, w;q;z)| ≤(−|w|;q)∞(−|z|;q)∞. Finally, for alln ∈Nwe have

(q¹⁻ⁿ;q)∞1φ1(0, q¹⁻ⁿ;q;z) = (−z)ⁿqⁿ⁽ⁿ⁻¹⁾² (q¹⁺ⁿ;q)∞1φ1(0, q¹⁺ⁿ;q;qⁿz).

Proof. See [10].

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Now we introduce the following functional spaces:

R^q ={∓qⁿ, n∈Z}, R⁺q ={qⁿ, n∈Z}.

Let D_q, C_q,0 and C_q,b denote the spaces of even smooth functions defined on Rq continuous at0, which are respectively with compact support, vanishing at infinity and bounded. These spaces are equipped with the topology of uniform convergence, and by L_q,p,v the space of even functions f defined on Rq such that

kfkq,p,v = Z ∞

0

|f(x)|^px^2v+1dqx ¹_p

<∞.

We denote by Sq the q−analogue of the Schwartz space of even function f defined on Rq such thatD^k_qf is continuous at0, and for alln ∈ Nthere isC_n such that

|D_q^kf(x)| ≤ C_n

(1 +x²)ⁿ, ∀k ∈N,∀x∈R⁺q.

Ar the end of this section we introduce theq−Bessel operator as follows

∆_q,vf(x) = 1 x²

f(q⁻¹x)−(1 +q^2v)f(x) +q^2vf(qx) . Proposition 1.2. Given two functionsf andg inL_q,2,v such that

∆_q,vf,∆_q,vg ∈ L_q,2,v

then Z ∞

0

∆q,vf(x)g(x)x^2v+1dqx= Z ∞

0

f(x)∆q,vg(x)x^2v+1dqx.

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2. The Normalized Hahn-Exton q−Bessel Function

The normalized Hahn-Extonq−Bessel function of ordervis defined as j_v(x, q) = (q, q)∞

(q^v+1, q)_∞x^−vJ_v⁽³⁾(x, q) =₁φ₁(0, q^v+1, q, qx²), <(v)>−1, whereJv⁽³⁾(·, q)is the Hahn-Extonq−bessel function, (see [12]).

Proposition 2.1. The function

x7→j_v(λx, q²), is a solution of the followingq−difference equation

∆_q,vf(x) = −λ²f(x) Proof. See [9].

In the following we put

c_q,v = 1

1−q · (q^2v+2, q²)∞

(q², q²)_∞ . Proposition 2.2. Letn, m∈Zandn 6=m, then we have

c²_q,v Z ∞

0

j_v(qⁿx, q²)j_v(q^mx, q²)x^2v+1d_qx= q^−2n(v+1) 1−q δ_nm. Proof. See [10].

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Proposition 2.3.

|j_v(qⁿ, q²)| ≤ (−q²;q²)∞(−q^2v+2;q²)∞

(q^2v+2;q²)∞

( 1 if n ≥0, qⁿ²^+(2v+1)n if n < 0.

Proof. Use Proposition1.1.

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3. q−Bessel Fourier Transform

Theq−Bessel Fourier transformF_q,vis defined as follows F_q,v(f)(x) = c_q,v

Z ∞

0

f(t)j_v(xt, q²)t^2v+1d_qt.

Proposition 3.1. Theq−Bessel Fourier transform F_q,v :L_q,1,v→ C_q,0, satisfying

kF_q,v(f)kC_q,0 ≤B_q,vkfk_q,1,v, where

Bq,v = 1 1−q

(−q²;q²)_∞(−q^2v+2;q²)_∞ (q²;q²)∞

. Proof. Use Proposition2.3.

Theorem 3.2. Givenf ∈ L_q,1,vthen we have

F_q,v² (f)(x) =f(x), ∀x∈R⁺q. Iff ∈ L_q,1,vandF_q,v(f)∈ L_q,1,vthen

kF_q,v(f)k_q,2,v=kfk_q,2,v.

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Proof. Lett, y∈R⁺q, we put

δ_q,v(t, y) =

( 1

(1−q)t^2v+2 if t=y,

0 if t6=y.

It is not hard to see that Z ∞

0

f(t)δ_q,v(t, y)t^2v+1d_qt =f(y).

By Proposition2.2, we can write

c²_q,v Z ∞

0

j_v(yx, q²)j_v(tx, q²)x^2v+1d_qx=δ_q,v(t, y), ∀t, y∈R⁺q, which leads to the result.

Corollary 3.3. The transformation

F_q,v :S_q → S_q, is an isomorphism, and

F_q,v⁻¹ =Fq,v.

Proof. The result is deduced from properties of the spaceS_q.

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4. q−Bessel Translation Operator

We introduce theq−Bessel translation operator as follows:

T_q,x^v f(y) =c_q,v Z ∞

0

F_q,v(f)(t)j_v(xt, q²)j_v(yt, q²)t^2v+1d_qt,

∀x, y ∈R⁺q,∀f ∈ Lq,1,v. Proposition 4.1. For any functionf ∈ Lq,1,vwe have

T_q,x^v f(y) =T_q,y^v f(x), and

T_q,x^v f(0) =f(x).

Proposition 4.2. For allx, y ∈R⁺q, we have

T_q,x^v j_v(λy, q²) =j_v(λx, q²)j_v(λy, q²).

Proposition 4.3. Letf ∈ Lq,1,vthen T_q,x^v f(y) =

Z ∞

0

f(z)D_v(x, y, z)z^2v+1d_qz, where

D_v(x, y, z) = c²_q,v Z ∞

0

j_v(xt, q²)j_v(yt, q²)j_v(zt, q²)t^2v+1d_qt.

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Proof. Indeed, T_q,x^v f(y)

=c_q,v Z ∞

0

F_q,v(f)(t)j_v(xt, q²)j_v(yt, q²)t^2v+1d_qt

=c_q,v Z ∞

0

c_q,v

Z ∞

0

f(z)j_v(zt, q²)z^2v+1d_qt

j_v(xt, q²)j_v(yt, q²)t^2v+1d_qt

= Z ∞

0

f(z)

c²_q,v Z ∞

0

j_v(xt, q²)j_v(yt, q²)j_v(zt, q²)t^2v+1d_qt

z^2v+1d_qz, which leads to the result.

Proposition 4.4.

z→∞lim D_v(x, y, z) = 0 and

(1−q)X

s∈Z

q^(2v+2)sD_v(x, y, q^s) = 1

Proof. To prove the first relation use Proposition 3.1. The second identity is deduced from Proposition4.2: iff = 1thenT_q,x^v f = 1.

Proposition 4.5. Givenf ∈ S_q then T_q,x^v f(y) =

∞

X

n=0

qⁿ⁽ⁿ⁺¹⁾

(q², q²)_n(q^2v+2, q²)_ny²ⁿ∆ⁿ_q,vf(x).

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Proof. By the use of Proposition2.1and the fact that

∆ⁿ_q,vf(x) = (−1)ⁿc_q,v Z ∞

0

F_q,v(f)(t)t²ⁿj_v(xt, q²)t^2v+1d_qt.

Proposition 4.6. Ifv =−¹₂ then Dv(q^m, q^r, q^k) = q2(r−m)(k−m)−m

(1−q)(q;q)∞

(q^2(r−m)+1;q)∞1φ1(0, q^2(r−m)+1, q;q^2(k−m)+1).

Proof. Indeed

∆ⁿ_q,v = q^−n(n+1) x²ⁿ

n

X

k=−n

2n k+n

q

(−1)^k+nq(k+n)(k+n+1)

2 −2kn

Λ^k_q,

and use Proposition4.5.

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5. q−Convolution Product

In harmonic analysis the positivity of the translation operator is crucial. It plays a central role in establishing some useful results, such as the property of the convolution product. Thus it is natural to investigate when this property holds forT_q,x^v . In the following we put

Q_v ={q∈[0,1], T_q,x^v is positive for all x∈R⁺q}.

Recall thatT_q,x^v is said to be positive ifT_q,x^v f ≥0forf ≥0.

Proposition 5.1. Ifv =−¹₂ then

Q_v = [0, q₀], whereq₀is the first zero of the following function:

q 7→₁φ₁(0, q, q, q).

Proof. The operatorT_q,x^v is positive if and only if

D_v(x, y, q^s)≥0, ∀x, y, q^s ∈R⁺q. We replace ^x_y byq^r, and we can chooser∈N, because

T_q,x^v f(y) =T_q,y^v f(x), thus we get

(q^1+2s, q)∞1φ1(0, q^1+2s, q, q^1+2r) =

∞

X

n=0

Bn(s, r), ∀r, s∈N,

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where

B_n(s, r) =

2n

Y

i=1

q^2r+i 1−qⁱ

∞

Y

i=2n+2

(1−q^2s+i)

×

(1−q^2s+2n+1)− q^2r+2n+1 1−q²ⁿ⁺¹

, ∀n ∈N^∗, and

B0(s, r) =

∞

Y

i=2

(1−q^2s+i)

(1−q^2s+1)− q^2r+1 1−q

, which leads to the result.

In the rest of this work we chooseq∈Q_v. Proposition 5.2. Givenf ∈ L_q,1,vthen

Z ∞

0

T_q,x^v f(y)y^2v+1d_qy = Z ∞

0

f(y)y^2v+1d_qy.

Theq−convolution product of both functionsf, g∈ Lq,1,vis defined by f∗_qg(x) =c_q

Z ∞

0

T_q,x^v f(y)g(y)y^2v+1d_qy.

Proposition 5.3. Given two functionsf, g∈ L_q,1,v then f ∗_qg ∈ L_q,1,v,

and

Fq,v(f∗qg) =Fq,v(f)Fq,v(g).

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Proof. We have

kf∗qgkq,1,v ≤ kfkq,1,vkgkq,1,v. On the other hand

Fq,v(f ∗qg)(λ) = Z ∞

0

Z ∞

0

f(x)T_q,y^v jv(λx, q²)x^2v+1dqx

g(y)y^2v+1dqy

=F_q,v(f)(λ) F_q,v(g)(λ).

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6. q−Heat Semigroup

Theq−heat semigroup is defined by:

P_q,t^v f(x) =G^v(·, t, q²)∗_qf(x)

=c_q,v Z ∞

0

T_q,x^v G^v(y, t, q²)f(y)y^2v+1d_qy, ∀f ∈ L_q,1,v. G^v(·, t, q²)is theq−Gauss kernel ofP_q,t^v

G^v(x, t, q²) = (−q^2v+2t,−q^−2v/t;q²)∞

(−t,−q²/t;q²)∞

e

−q^−2v t x², q²

. ande(·, q)theq-exponential function defined by

e(z, q) =

∞

X

n=0

zⁿ

(q, q)_n = 1 (z;q)∞

, |z|<1.

Proposition 6.1. Theq−Gauss kernelG^v(·, t, q²)satisfying F_q,v

G^v(·, t, q²) (x) = e(−tx², q²), and

F_q,v

e(−ty², q²) (x) =G^v(x, t, q²).

Proof. In [5], the Ramanujan identity was proved

X

s∈Z

z^s (bq^2s, q²)∞

=

bz,^q_bz², q², q²

∞

b, z,^q_b², q²

∞

,

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which implies Z ∞

0

e(−ty², q²)y²ⁿy^2v+1d_qy= (1−q)X

s

(q^2n+2v+2)^s (−tq^2s, q²)∞

= (1−q)

−tq^2n+2v+2,−^q^−2n−2v_t , q², q²

∞

−t, q^2n+2v+2,−^q_t², q²

∞

.

The following identity leads to the result

(a, q²)∞ = (a, q²)n(q²ⁿa, q²)∞, and

(aq⁻²ⁿ, q²)∞= (−1)ⁿq⁻ⁿ²⁺ⁿ a

q² n

q² a , q²

n

(a, q²)∞.

Proposition 6.2. For any functionsf ∈ S_q, we have P_q,t^v f(x) = e(t∆q,v, q²)f(x).

Proof. Indeed, if

c_q,v Z ∞

0

G^v(y, t, q²)y²ⁿy^2v+1d_qy= (q^2v+2, q²)_nq^−n(n+n)tⁿ,

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then

P_q,t^v f(x) =

∞

X

n=0

qⁿ⁽ⁿ⁺¹⁾ (q², q²)_n(q^2v+2, q²)_n

×

c_q,v Z ∞

0

G^v(y, t, q²)y²ⁿy^2v+1d_qy

∆ⁿ_q,vf(x).

Theorem 6.3. Forf ∈ Lq,p,vand1≤p < ∞, we have kP_q,t^v fk_q,p,v ≤ kfk_q,p,v. Proof. Ifp= 1then

kP_q,t^v fk_q,1,v ≤ kG^v(·, t, q²)k_q,1,vkfk_q,1,v =kfk_q,1,v. Now letp >1and we consider the following function

g :y7→T_q,x^v G^v(y, t;q²).

In addition P_q,t^v f

p

q,p≤c^p_q,v Z ∞

0

Z ∞

0

|f(y)g(y)|y^2v+1dqy p

x^2v+1dqx.

By the use of the Hölder inequality and the fact that kG^v(·, t, q²)k_q,1,v = _c¹

q,v, the result follows immediately.

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7. q−Wiener Algebra

Foru∈ Lq,1,vandλ∈R⁺q, we introduce the following function u_λ :x7→ 1

λ^2v+2ux λ

. Proposition 7.1. Givenu∈ L_q,1,v such that

Z ∞

0

u(x)x^2v+1d_qx= 1, then we have

λ→0lim Z ∞

0

f(x)u_λ(x)x^2v+1d_qx=f(0), ∀f ∈ C_q,b. Corollary 7.2. The following function

G^v_λ :x7→c_q,vG^v(x, λ², q²), checks the conditions of the preceding proposition.

Theorem 7.3. Givenf ∈ Lq,1,v∩ Lq,p,v,1≤p <∞andfλ defined by f_λ(x) =c_q

Z ∞

0

F_q,v(f)(y)e(−λ²y², q²)j_v(xy, q²)y^2v+1d_qy.

then we have

λ→0limkf−f_λk_q,p,v = 0.

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Proof. We have

f ∗_qG^v_λ(x) =c_q,v Z ∞

0

F_q,v(f)(t)e(−λ²t², q²)j_v(tx, q²)t^2v+1d_qt.

In addition, for allε >0, there exists a functionh∈ L_q,p,vwith compact support in[q^k, q^−k]such that

kf−hk_q,p,v < ε, however

kG^v_λ∗_qf−fk_q,p,v ≤ kG^v_λ∗_q(f−h)k_q,p,v+kG^v_λ∗_qh−hk_q,p,v+kf−hk_q,p,v. By Theorem6.3we get

kG^v_λ∗q(f−h)kq,p,v ≤ kf −hkq,p,v. Now, we will prove that

λ→0limkG^v_λ ∗qh−hkq,p,v = 0.

Indeed, by the use of Corollary7.2we get

λ→0lim Z 1

0

|G^v_λ∗_qh(x)−h(x)|^px^2v+1d_qx= 0.

On the other hand the following function is decreasing on the interval[1,∞[:

u7→u^2v+2G^v(u).

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Ifλ <1, then we deduce that

T_q,q^v iG^v_λ(x)≤T_q,q^v iG(x).

We can use the dominated convergence theorem to prove that

λ→0lim Z ∞

1

|G^v_λ∗_qh(x)−h(x)|^px^2v+1d_qx= 0.

Corollary 7.4. Givenf ∈ Lq,1,vthen f(x) = c_q,v

Z ∞

0

F_q,v(f)(y)j_v(xy, q²)y^2v+1d_qy, ∀x∈R⁺q. Proof. The result is deduced by Theorem7.3and the following relation

(1−q)x^2v+2|f(x)−f_λ(x)| ≤ kf −f_λk_q,1,v ∀x∈R⁺q.

Now we attempt to study theq−Wiener algebra denoted by Aq,v ={f ∈ Lq,1,v, Fq,v(f)∈ Lq,1,v}. Proposition 7.5. For1≤p≤ ∞, we have

1. A_q,v ⊂ L_q,p,v and A_q,v =L_q,p,v.

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2. A_q,v ⊂ C_q,0 and A_q,v =C_q,0.

Proof. 1. Given h ∈ L_q,p,v with compact support, and we put h_n = h∗_qG^v_qn. The functionh_n ∈ A_q,vand by Theorem7.3we get

n→∞lim kh−h_nk_q,p,v = 0.

2. Iff ∈ C_q,0, then there existh ∈ C_q,0 with compact support on[q^k, q^−k], such that

kf−hk_C_q,0 < ε, and by Corollary7.4we prove that

n→∞lim

"

sup

x∈R⁺q

|h(x)−h_n(x)|

#

= 0.

Theorem 7.6. Forf ∈ L_q,2,v∩ L_q,1,v, we have kFq,v(f)kq,2,v=kfkq,2,v. Proof. We put

f_n =f ∗_qG^v_qn, which implies

F_q,v(f_n)(t) = e(−q²ⁿt², q²)F_q,v(f)(t),

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by Corollary7.4we get

f_n(x) =c_q Z ∞

0

F_q,v(f_n)(t)j_v(tx, q²)t^2v+1d_qt.

On the other hand Z ∞

0

f(x)f_n(x)x^2v+1d_qx= Z ∞

0

F_q,v(f)(x)F_q,v(f_n)(x)x^2v+1d_qx.

Theorem7.3implies

n→∞lim Z ∞

0

F_q,v(f)(x)²e(−q²ⁿx², q²)x^2v+1d_qx=kfk²_q,2,v.

The sequencee(−q²ⁿx², q²)is increasing. By the use of the Fatou-Beppo-Levi theorem we deduce the result.

Theorem 7.7.

1. Theq−cosine Fourier transformF_q,v possesses an extension U :L_q,2,v→ L_q,2,v.

2. Forf ∈ L_q,2,v, we have

kU(f)k_q,2,v =kfk_q,2,v. 3. The applicationU is bijective and

U⁻¹ =U.

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Proof. Let the maps

u:A_q,v → A_q,v, f 7→ F_q,v(f).

Theorem3.2implies

ku(f)k_q,2,v=kfk_q,2,v.

The mapuis uniformly continuous, with values in complete spaceL_q,2,v. It has a prolongationU onAq,v =Lq,2,v.

Proposition 7.8. Given1< p ≤2and ¹_p+_p¹0 = 1, iff ∈ L_q,p,v, thenF_q,v(f)∈ L_q,p⁰_,v,

kF_q,v(f)k_q,p⁰_,v ≤B_p,q,vkfk_q,p,v, where

B_p,q,v =B⁽

2 p−1) q,v .

Proof. The result is a consequence of Proposition 3.1, Theorem 7.7 and the Riesz-Thorin theorem, see [13].

As an immediate consequence of Proposition 7.8, we have the following theorem:

Theorem 7.9. Given1< p, p⁰, r≤2and 1

p+ 1

p⁰ −1 = 1 r, iff ∈ L_q,p,vandg ∈ L_q,p⁰_,v, then

f∗_qg ∈ L_q,r,v, and

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kf∗_qgk_q,r,v ≤B_q,p,vB_q,p⁰_,vB_q,r⁰_,vkfk_q,p,vkgk_q,p⁰_,v,

where 1

r + 1 r⁰ = 1.

Proof. We can write

f∗_qg =F_q,v{F_q,v(f)F_q,v(g)},

the use of Proposition7.8and the Hölder inequality leads to the result.

Now we are in a position to establish the hypercontractivity of theq−heat semigroupP_q,t^v . For more information about this notion, the reader can consult ([1,2,3]).

Proposition 7.10. Forf ∈ Lq,p⁰,vandt∈R⁺q, we have

kP_q,t^v fk_q,p,v ≤B_q,p⁰_,vB_q,p₁_,vc(r, q, v)t⁻^v+1^r kfk_q,p⁰_,v, where

1< p⁰ < p ≤2, 1 p + 1

p1

= 1, 1 r = 1

p⁰ −1 p, and

c(r, q, v) =ke(−x², q²)k_q,r,v. Proof. The result is deduced by the following relations

F_q,v

G^v(·, t, q²) (x) = e(−tx², q²), and

kF_q,v

G^v(·, t, q²) k_q,r,v =c(r, q, v)t⁻^v+1^r .

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8. q−Central Limit Theorem

In this section we study the analogoue of the well known central limit theorem with the aid of theq−Bessel Fourier transform.

For this, we consider the setM⁺_q of positive and bounded measures onR⁺q. Theq-cosine Fourier transform ofξ ∈ M⁺_q is defined by

F_q,v(ξ)(x) = Z ∞

0

j_v(tx, q²)t^2v+1d_qξ(t).

Theq−convolution product of two measuresξ, ρ ∈ M⁺_q is given by ξ∗_qρ(f) =

Z ∞

0

T_q,x^v f(t)t^2v+1d_qξ(x)d_qρ(t), and we have

F_q,v(ξ∗_qρ) =F_q,v(ξ)F_q,v(ρ).

We begin by showing the following result

Proposition 8.1. Forf ∈ A_q,vandξ∈ M⁺_q, we have Z ∞

0

f(x)x^2v+1dqξ(x) =cq,v

Z ∞

0

Fq,v(f)(x)Fq,v(ξ)(x)x^2v+1dqx.

As a direct consequence we may state Corollary 8.2. Givenξ, ξ⁰ ∈ M⁺_q such that

F_q,v(ξ) =F_q,v(ξ⁰), thenξ=ξ⁰.

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Proof. By Proposition8.1, we have Z ∞

0

f(x)x^2v+1d_qξ(x) = Z ∞

0

f(x)x^2v+1d_qξ⁰(x), ∀f ∈ A_q,v. from the assertion (2) of Proposition7.5, we conclude thatξ=ξ⁰.

Theorem 8.3. Let(ξ_n)_n≥0 be a sequence of probability measures ofM⁺_q such that

n→∞lim F_q,v(ξ_n)(t) =ψ(t),

then there existsξ ∈ M⁺_q such that the sequenceξ_nconverges strongly toward ξ, and

Fq,v(ξ) =ψ.

Proof. We consider the mapI_ndefined by I_n(u) =

Z ∞

0

u(x)x^2v+1d_qξ_n(x), ∀f ∈ C_q,0. By the following inequality

|I_n(u)| ≤ kuk_C_q,0, and by Proposition8.1, we get

I_n(f) =c_q,v Z ∞

0

F_q,v(f)(x)F_q,v(ξ_n)(x)x^2v+1d_qx, ∀f ∈ A_q,v,

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which implies

n→∞lim I_n(f) = Z ∞

0

F_q,v(f)(x)ψ(x)x^2v+1d_qx, ∀f ∈ A_q,v.

On the other hand, by assertion (2) of Proposition 7.5, and by the use of the Ascoli theorem (see [11]):

Consider a sequence of equicontinuous linear forms onC_q,0which converge on a dense partAq,v then converge on the entireCq,0. We get

n→∞lim In(u) = Z ∞

0

Fq,v(u)(x)ψ(x)x^2v+1dqx, ∀u∈ Cq,0. Finally there existξ ∈ M⁺_q such that

n→∞lim Z ∞

0

u(x)x^2v+1d_qξ_n(x) = Z ∞

0

u(x)x^2v+1d_qξ(x), ∀u∈ C_q,0. On the other hand

Fq,v(Aq,v) =Aq,v, and

Z ∞

0

Fq,v(f)(x)Fq,v(ξ)(x)dqx= Z ∞

0

Fq,v(f)(x)ψ(x)x^2v+1dqx, ∀f ∈ Aq,v, which implies

F_q,v(ξ) =ψ.

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Proposition 8.4. Givenξ∈ M⁺_q, and supposing that σ =

Z ∞

0

t²t^2v+1d_qξ(t)<∞, then

F_q,v(ξ)(x) = 1− q²σ

(q², q²)₁(q^2v+2, q²)₁x²+o(x²).

Proof. We write

j_v(tx, q²) = 1− q²t²

(q², q²)₁(q^2v+2, q²)₁x²+x²θ(tx)t², where

limx→0θ(x) = 0, then

F_q,v(ξ)(x) = 1− q²σ

(q², q²)₁(q^2v+2, q²)₁x²+ Z ∞

0

t²θ(tx)t^2v+1d_qξ(t)

x².

Now we are in a position to present theq−central limit theorem.

Theorem 8.5. Let (ξ_n)n≥0 be a sequence of probability measures of M⁺_q of total mass1, satisfying

n→∞lim nσ_n =σ, where σ_n= Z ∞

0

t²t^2v+1d_qξ_n(t),

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and

n→∞lim neσ_n= 0, where σe_n = Z ∞

0

t⁴

1 +t²t^2v+1d_qξ_n(t), thenξ_n^∗nconverge strongly toward a measureξdefined by

d_qξ(x) =c_q,vF_q,v

e⁻

q2σ

(q2,q2)1(q2v+2,q2)1t²

(x)d_qx.

Proof. We have

F_q,v(ξ_n^∗n) = (F_q,v(ξ_n))ⁿ, and

F_q,v(ξ_n)(x) = 1− q²σ_n

(q², q²)₁(q^2v+2, q²)₁x²+θ_n(x)x², where

θ_n(x) = Z ∞

0

t²θ(tx)t^2v+1d_qξ_n(t).

Consequently

(Fq,v(ξn))ⁿ(x) = exp

nlog

1− q²σ_n

(q², q²)₁(q^2v+2, q²)₁x²+θn(x)x²

. By the following inequality

|t²θ(tx)| ≤C_x t⁴

1 +t², ∀t ∈R⁺q, whereC_x is some constant, the result follows immediately.

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References

[1] K.I. BABENKO, An inequality in the theory of Fourier integrals, Izv.

Akad. Nauk SSSR, 25 (1961), English transl., Amer. Math. Soc.

[2] W. BECKNER, Inequalities in Fourier analysis, Ann.of Math.,(2), 102 (1975), 159–182.

[3] A. FITOUHI, Inégalité de Babenko et inégalité logarithmique de Sobolev pour l’opérateur de Bessel, C.R. Acad. Sci. Paris, 305(I) (1987), 877–880.

[4] G. GASPER AND M. RAHMAN, Basic hypergeometric series, Ency- clopedia of Mathematics and its Applications, 35, Cambridge University Press, 1990.

[5] M.E.H. ISMAIL, A simple proof of Ramanujan’s ₁ψ₁ sum, Proc. Amer.

Math. Soc., 63 (1977), 185–186.

[6] F.H. JACKSON, Onq-Functions and a certain difference operator, Trans- actions of the Royal Society of London, 46 (1908), 253–281.

[7] F.H. JACKSON, On a q-definite integral, Quarterly Journal of Pure and Application Mathematics, 41 (1910), 193–203.

[8] T.H. KOORNWINDER, Special functions and q-commuting variables, in Special Functions, q-Series and Related Topics, M. E. H. Ismail, D. R.

Masson and .Rahman (eds), Fields Institue Communications 14, American Mathematical Society, 1997, pp. 131–166.

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[9] H.T. KOELINK AND R.F. SWARTTOUW, On the zeros of the Hahn- Exton q-Bessel function and associated q-Lommel polynomials, Journal of Mathematical Analysis and Applications, 186(3) (1994), 690–710.

[10] T.H. KOORNWINDERANDR.F. SWARTTOUW, Onq-Analogues of the Hankel and Fourier transform, Trans. A.M.S., 1992, 333, 445–461.

[11] L. SCHWARTZ, Analye Hilbertienne, Hermann Paris-Collection Méth- ode, 1979.

[12] R.F. SWARTTOUW, The Hahn-Exton q-Bessel functions, PhD Thesis, The Technical University of Delft, 1992.

[13] G.O. THORIN, Kungl.Fysiogr.söllsk.i Lund Förh, 8 (1938), 166–170.