With applications and finally we enunciate aq−Bessel version of the central limit theorem

http://jipam.vu.edu.au/

Volume 7, Issue 5, Article 171, 2006

INEQUALITIES IN q−FOURIER ANALYSIS

LAZHAR DHAOUADI, AHMED FITOUHI, AND J. EL KAMEL ECOLEPRÉPARATOIRE D’INGÉNIEUR

BIZERTE, TUNISIA. lazhardhaouadi@yahoo.fr FACULTÉ DESSCIENCES DETUNIS

1060 TUNIS, TUNISIA. Ahmed.Fitouhi@fst.rnu.tn

Received 17 April, 2006; accepted 28 July, 2006 Communicated by H.M. Srivastava

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

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

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

1. INTRODUCTION ANDPRELIMINARIES

In introducingq−Bessel Fourier transforms, theq−Bessel translation operator and theq−convolution product we shall use the standard conventional notation as described in [4]. For further detailed

information on q−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)ⁿq^n(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).

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

115-06

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)ⁿq^n(n−1)2 (q¹⁺ⁿ;q)∞1φ₁(0, q¹⁺ⁿ;q;qⁿz).

Proof. See [10].

Now we introduce the following functional spaces:

Rq ={∓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 at 0, which are respectively with compact support, vanishing at infinity and bounded. These spaces are equipped with the topology of uniform convergence, and byL_q,p,v the space of even functionsf defined onRqsuch that

kfk_q,p,v = Z ∞

0

|f(x)|^px^2v+1d_qx _p¹

<∞.

We denote byS_q theq−analogue of the Schwartz space of even functionf defined onRqsuch thatD^k_qf is continuous at0, and for alln ∈Nthere isC_nsuch 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 andginL_q,2,vsuch that

∆_q,vf,∆_q,vg ∈ L_q,2,v

then Z ∞

0

∆_q,vf(x)g(x)x^2v+1d_qx= Z ∞

0

f(x)∆_q,vg(x)x^2v+1d_qx.

2. THENORMALIZEDHAHN-EXTONq−BESSEL FUNCTION

The normalized Hahn-Extonq−Bessel function of orderv is 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].

Proposition 2.3.

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

( 1 if n≥0, q^n2+(2v+1)n if n <0.

Proof. Use Proposition 1.1.

3. q−BESSELFOURIERTRANSFORM

Theq−Bessel Fourier transformF_q,v is defined as follows Fq,v(f)(x) = cq,v

Z ∞

0

f(t)jv(xt, q²)t^2v+1dqt.

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

B_q,v = 1 1−q

(−q²;q²)∞(−q^2v+2;q²)∞

(q²;q²)∞

.

Proof. Use Proposition 2.3.

Theorem 3.2. Givenf ∈ L_q,1,v then we have

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

kF_q,v(f)k_q,2,v =kfk_q,2,v. Proof. Lett, y ∈R⁺q, we put

δ_q,v(t, y) =

( 1

(1−q)t^2v+2 if t =y,

0 if t 6=y.

It is not hard to see that

Z ∞

0

f(t)δ_q,v(t, y)t^2v+1d_qt=f(y).

By Proposition 2.2, 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 =F_q,v.

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

4. q−BESSELTRANSLATION OPERATOR

We introduce theq−Bessel translation operator as follows:

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

Z ∞

0

Fq,v(f)(t)jv(xt, q²)jv(yt, q²)t^2v+1dqt, ∀x, y ∈R⁺q,∀f ∈ Lq,1,v. Proposition 4.1. For any functionf ∈ L_q,1,v we 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).

Proof. Use Proposition 2.2.

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

Z ∞

0

f(z)Dv(x, y, z)z^2v+1dqz, 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.

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

jv(xt, q²)jv(yt, q²)jv(zt, q²)t^2v+1dqt

z^2v+1dqz,

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

Proposition 4.2: iff = 1thenT_q,x^v f = 1.

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

∞

X

n=0

qⁿ⁽ⁿ⁺¹⁾

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

Proof. By the use of Proposition 2.1 and 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

D_v(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 Proposition 4.5.

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

Qv ={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

Qv = [0, q0], whereq₀ is the first zero of the following function:

q7→₁φ₁(0, q, q, q).

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

Dv(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

B_n(s, r), ∀r, s∈N, 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,v then 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 ∈ L_q,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,vthen f ∗_qg ∈ L_q,1,v, and

F_q,v(f∗_qg) = F_q,v(f)F_q,v(g).

Proof. We have

kf ∗_qgk_q,1,v ≤ kfk_q,1,vkgk_q,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).

6. q−HEATSEMIGROUP

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

Fq,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

∞

,

which implies Z ∞

0

e(−ty², q²)y²ⁿy^2v+1dqy= (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, 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 ∈ L_q,p,vand1≤p < ∞, we have

kP_q,t^v fkq,p,v ≤ kfkq,p,v. Proof. Ifp= 1then

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

g :y 7→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+1d_qy p

x^2v+1d_qx.

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

q,v, the result follows

immediately.

7. q−WIENERALGEBRA

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

λ^2v+2ux λ

. Proposition 7.1. Givenu∈ L_q,1,vsuch 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→cq,vG^v(x, λ², q²), checks the conditions of the preceding proposition.

Proof. Use Proposition 6.1.

Theorem 7.3. Givenf ∈ L_q,1,v∩ L_q,p,v,1≤p < ∞andf_λ 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 −fλkq,p,v = 0.

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 all ε > 0, there exists a function h ∈ L_q,p,v with compact support in[q^k, q^−k] such that

kf −hk_q,p,v < ε, however

kG^v_λ∗qf −fkq,p,v≤ kG^v_λ∗q(f−h)kq,p,v+kG^v_λ∗qh−hkq,p,v+kf −hkq,p,v. By Theorem 6.3 we get

kG^v_λ ∗_q(f −h)k_q,p,v≤ kf −hk_q,p,v. Now, we will prove that

λ→0limkG^v_λ∗_qh−hk_q,p,v = 0.

Indeed, by the use of Corollary 7.2 we 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).

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 ∈ L_q,1,v then

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 Theorem 7.3 and 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

A_q,v ={f ∈ L_q,1,v, F_q,v(f)∈ L_q,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. (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 ∗_q G^v_qn. The function hn ∈ Aq,vand by Theorem 7.3 we 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_Cq,0 < ε,

and by Corollary 7.4 we 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

kF_q,v(f)k_q,2,v =kfk_q,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), by Corollary 7.4 we 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)fn(x)x^2v+1dqx= Z ∞

0

Fq,v(f)(x)Fq,v(fn)(x)x^2v+1dqx.

Theorem 7.3 implies

n→∞lim Z ∞

0

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

The sequence e(−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,vpossesses an extension U :L_q,2,v → L_q,2,v.

(2) Forf ∈ L_q,2,v, we have

kU(f)kq,2,v =kfkq,2,v. (3) The applicationU is bijective and

U⁻¹ =U.

Proof. Let the maps

u:A_q,v → A_q,v, f 7→ F_q,v(f).

Theorem 3.2 implies

ku(f)kq,2,v =kfkq,2,v.

The mapuis uniformly continuous, with values in complete spaceL_q,2,v. It has a prolongation

U onA_q,v =L_q,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

Bp,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 theo-

rem, 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,v andg ∈ L_q,p⁰_,v, then

f∗_qg ∈ L_q,r,v, and

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 Proposition 7.8 and the Hölder inequality leads to the result.

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

Proposition 7.10. Forf ∈ L_q,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+1r 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+1r .

8. q−CENTRALLIMIT 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 set M⁺_q of positive and bounded measures on R⁺q. The q-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 ∈ Aq,vandξ ∈ M⁺_q, we have Z ∞

0

f(x)x^2v+1d_qξ(x) =c_q,v Z ∞

0

F_q,v(f)(x)F_q,v(ξ)(x)x^2v+1d_qx.

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.

Proof. By Proposition 8.1, 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 Proposition 7.5, 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 towardξ, and F_q,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

|In(u)| ≤ kukCq,0, and by Proposition 8.1, 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, which implies

n→∞lim In(f) = Z ∞

0

Fq,v(f)(x)ψ(x)x^2v+1dqx, ∀f ∈ Aq,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,0 which converge on a dense part A_q,v then converge on the entireC_q,0. We get

n→∞lim I_n(u) = Z ∞

0

F_q,v(u)(x)ψ(x)x^2v+1d_qx, ∀u∈ C_q,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

F_q,v(f)(x)F_q,v(ξ)(x)d_qx= Z ∞

0

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

F_q,v(ξ) = ψ.

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

x→0limθ(x) = 0, then

Fq,v(ξ)(x) = 1− q²σ

(q², q²)₁(q^2v+2, q²)₁x²+ Z ∞

0

t²θ(tx)t^2v+1dqξ(t)

x².

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

Theorem 8.5. Let(ξ_n)n≥0be a sequence of probability measures ofM⁺_q of total mass1, satis- fying

n→∞lim nσ_n=σ, where σ_n = Z ∞

0

t²t^2v+1d_qξ_n(t), and

n→∞lim neσn= 0, where eσn= Z ∞

0

t⁴

1 +t²t^2v+1dqξn(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

θn(x) = Z ∞

0

t²θ(tx)t^2v+1dqξn(t).

Consequently

(F_q,v(ξ_n))ⁿ(x) = exp

nlog

1− q²σn

(q², q²)₁(q^2v+2, q²)₁x²+θ_n(x)x²

. By the following inequality

|t²θ(tx)| ≤C_x t⁴

1 +t², ∀t∈R⁺q,

whereC_xis some constant, the result follows immediately.

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