() CUBO A Mathematical Journal Vol.17, No¯ 02, (123–141). June 2015 Calderón’s reproducing Formula For q-Bessel operator Belgacem Selmi Faculté des Sciences de Bizerte, Département de Mathématiques, 7021 Zarzouna, Tunisie. belgacem.selmi@fsb.rnu.tn ABSTRACT In this paper a Calderón-type reproducing formula for q-Bessel convolution is estab- lished using the theory of q-Bessel Fourier transform [13, 17], obtained in Quantum calculus. RESUMEN En este trabajo se prueba una fórmula de tipo Calderón para convolución q-Bessel, usando la teoŕıa de q-Bessel transformada de Fourier [13, 17], obtenida en cálculo cuántico. Keywords and Phrases: q-Calderon, q-Calculus, q-Bessel Convolution, q-Fourier Bessel trans- form, q-Measure. 2010 AMS Mathematics Subject Classification: 05A30, 33DXX, 44A15, 33D15. 124 Belgacem Selmi CUBO 17, 2 (2015) 1 Introduction Calderón’s formula [1] involving convolutions related to the Fourier transform is useful in obtaining reconstruction formula for wavelet transform, in decomposition of certain spaces and in character- ization of Besov spaces [6, 8, 10]. Calderón’s reproducing formula was also established for Bessel operator [4, 5]. This work is a continuation of a last work [9], and we establish formula for q-Bessel convolution for both functions and measures witch generalize the above one. In the classical case this formula is expressed for a suitable function f as follows: f(x) = ∫ ∞ 0 (gt ∗ ht ∗ f)(x) dt t , (1) where g, h ∈ L2(R) and gt(x) = 1 t g(x t ), ht(x) = 1 t h(x t ), t > 0 satisfying ∫ ∞ 0 ĝ(xt)ĥ(xt) dx x = 1, for all t ∈ R \ {0}, where ĝ and ĥ is the usual Fourier transform of g and h on R. If µ is a finite Borel measure on the real line R, identity (1) has natural generalization as follow f(x) = ∫ ∞ 0 (f ∗ µt)(x) dt t , (2) where µt is the dilated measure of µ under some restriction on µ, the L p-norm of (2) has proved in [2]. A general form of (2) has been investigated in [3]. In this paper we study similar questions when in (1) and (2) the classical convolution ∗ is replaced by the q-Bessel convolution ∗α,q on the half line generated by the q-Bessel operator defined by ∆q,αf(x) = 1 x2α+1 Dq [ x2α+1Dqf ] (q−1x). (3) In this paper we prove that, for ϕ and ψ ∈ L1α,q(Rq,+,dqσ(x)) satisfying ∫ ∞ 0 Fα,q(ϕ)(ξ)Fα,q(ψ)(ξ) dqξ ξ = 1 (4) we have f(x) = ∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t , f ∈ L1α,q(Rq,+,dqσ(x)). (5) where dqσ(x) = (1 + q)−α Γq2(α + 1) x2α+1dqx = bα,qx 2α+1dqx, ϕt(x) = 1 t2α+2 ϕ( x t ). In particular for ϕ ∈ L1α,q(Rq,+,dqσ(x)) such that ∫ ∞ 0 [Fα,q(ϕ)(ξ)] 2 dqξ ξ = 1, (6) CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 125 and for a suitable function f, put fε,δ(x) = ∫δ ε (f ∗α,q ϕt ∗α,q ϕt)(x) dqt t (7) then ‖fε,δ − f‖2,α,q −→ 0 as ε → 0 and δ → ∞. (8) In the case f ∈ L1α,q(Rq,+,dqσ(x)) such that Fα,qf ∈ L 1 α,q(Rq,+,dqσ(x)) one has lim ε → 0 δ → ∞ fε,δ(x) = f(x), x ∈ R. (9) Then we prove that for µ ∈ M ′ (Rq,+), such that the q-integral cµ,α,q = ∫ ∞ 0 Fα,q(µ)(λ) dqλ λ (10) is finite. Then for all f ∈ L2α,q(Rq,+,dqσ(x)), we have lim ε → 0 δ → ∞ fε,δ = cµ,α,qf. (11) where the limit is in L2α,q(Rq,+,dqσ(x)). And if µ ∈ M ′ (Rq,+) is such that the q-integral ∫ ∞ 0 |µ([0,y])| dqy y (12) is finite, for all f ∈ L2α,q(Rq,+,dqσ(x)) lim ε → 0 δ → ∞ fε,δ = cµ,α,qf, in L 2 α,q(Rq,+,dqσ(x)). (13) The outline of this paper is as follows: In Section 2, basic properties of q-Bessel transform on Rq of functions and bounded measure and its underlying q-convolution structure are called and introduced here. In Section 3, we give the first main result of the paper, the q-Calderon’s reproducing formula for functions. Section 4 is consecrate to establish the same result as in section 3 for finite measures. 2 Preliminaries In this section we recall some basic result in harmonic analysis related to the q-Bessel Fourier transform. Standard reference here is Gasper & Rahman [7]. 126 Belgacem Selmi CUBO 17, 2 (2015) For a,q ∈ C the q-shifted factorial (a;q)k is defined as a product of k factors: (a;q)k = (1 − a)(1 − aq)...(1 − aq k−1), k ∈ N∗; (a;q)0 = 1. (14) If |q| < 1 this definition remains meaningful for k = +∞ as a convergent infinite product: (a;q)∞ = ∞∏ k=0 (1 − aqk). (15) We also write (a1, ...,ar;q)k for the product of r q-shifted factorials: (a1, · · · ,ar;q)k = (a1;q)k...(ar;q)k (k ∈ N or k = ∞). (16) A q-hypergeometric series is a power series (for the moment still formal) in one complex variable z with power series coefficients which depend, apart from q, on r complex upper parameters a1, ...,ar and s complex lower parameters b1, ...,bs as follows: rϕs(a1, · · · ,ar;b1, · · · ,bs;q,x) = ∞∑ k=0 (a1, · · · ,ar;q)k (b1, · · · ,bs;q)k(q;q)k [(−1)kq k(k−1) 2 ] 1+s−rxk (for r,s ∈ N). 2.1 q-Exponential series eq(z) = 1ϕ0(0; −;q,z) = ∞∑ k=0 zk (q;q)k = 1 (z;q)∞ (| z |< 1) (17) Eq(z) = 0ϕ0(−; −;q,−z) = ∞∑ k=0 q 1 2 k(k−1)zk (q;q)k = (−z;q)∞ (z ∈ C). (18) 2.2 q-Derivative and q-Integral The q-derivative of a function f given on a subset of R or C is defined by: Dqf(x) = f(x) − f(qx) (1 − q)x (x 6= 0,q 6= 0), (19) where x and qx should be in the domain of f. By continuity we set (Dqf)(0) = f ′ (0) provided f ′ (0) exists. The q-shift operators are (Λqf)(x) = f(qx), (Λ −1 q f)(x) = f(q −1x). (20) CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 127 For a ∈ R \ {0} and a function f given on (0,a] or [a,0), we define the q-integral by ∫a 0 f(x)dqx = (1 − q)a ∞∑ n=0 f(aqn)qn, (21) provided the infinite sum converges absolutely (for instance if f is bounded). If F(a) is given by the left-hand side of (21) then DqF = f. The right-hand side of (21) is an infinite Riemann sum. For a q-integral over (0,∞) we define ∫ ∞ 0 f(x)dqx = (1 − q) +∞∑ −∞ f(qk)qk. (22) Note that for n ∈ Z and a ∈ Rq, we have ∫ ∞ 0 f(qnx)dqx = 1 qn ∫ ∞ 0 f(x)dqx, ∫a 0 f(qnx)dqx = 1 qn ∫aqn 0 f(x)dqx. (23) The q-integration by parts is given for suitable functions f and g by: ∫b a f(x)Dqg(x)dqx = [ f(x)g(x) ]b a − ∫b a Dqf(x)g(x)dqx. (24) The q-Logarithm logq is given by [19] logq x = ∫ dqx x = 1 − q logq logx. (25) For all a,b ∈ qZ,a < b logq(b/a) = (1 − q) ∑ k:a≤qk≤b 1. (26) The improper integral is defined in the following way ∫ ∞/A 0 f(x)dqx = (1 − q) +∞∑ −∞ f ( qn A ) qn A . (27) We remark that for n ∈ Z, we have ∫ ∞/qn 0 f(x)dqx = ∫ ∞ 0 f(x)dqx. (28) The following property holds for suitable function f ∫ ∞ 0 ∫x 0 f(x,y)dqydqx = ∫ ∞ 0 ∫ ∞ qy f(x,y)dqxdqy. (29) 128 Belgacem Selmi CUBO 17, 2 (2015) 2.3 The q-gamma function The q-gamma function is defined by [7, 16] Γq(z) = (q;q)∞ (qz;q)∞ (1 − q)1−z, 0 < q < 1;z 6= 0,−1,−2, ... (30) = ∫(1−q)−1 0 tz−1Eq(−(1 − q)qt)dqt, (Rez > 0) (31) moreover the q-duplication formula holds Γq(2z)Γq2( 1 2 ) = (1 + q)2z−1Γ2q(z)Γq2(z + 1 2 ). (32) 2.4 Some q-functional spaces We begin by putting Rq,+ = {+q k,k ∈ Z}, R̃q,+ = {+q k,k ∈ Z} ∪ {0} (33) and we denote by • L p α,q(Rq,+), p ∈ [1,+∞[, ( resp. L ∞ α,q(Rq,+) ) the space of functions f such that, ‖f‖p,α,q = ( ∫ ∞ 0 |f(x)|pdqσ(x)) 1 p < +∞. (34) (resp. ‖f‖∞,q = ess sup x∈Rq,+ |f(x)| < +∞). (35) • Sq,∗(Rq) the q-analogue of Schwartz space of even functions defined on Rq such that D k q,xf(x) is continuous in 0 for all k ∈ N and Nq,n,k(f) = sup x∈Rq |(1 + x2)nDkq,xf(x)| < +∞. (36) • The q-analogue of the tempered distributions is introduced in [12] as follow: (i) A q-distribution T in Rq is said to be tempered if there exists Cq > 0 and k ∈ N such that: |〈T,f〉| ≤ CqNq,n,k(f); f ∈ Sq,∗(Rq). (37) (ii) A linear form T: Sq,∗(Rq) −→ C is said continuous if there exist Cq > 0 and k ∈ N such that: |〈T,f〉| ≤ CqNq,n,k(f); f ∈ Sq,∗(Rq). (38) CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 129 • S ′ q,∗(Rq) the space of even q-tempered distributions in Rq. That is the topological dual of Sq,∗(Rq). • Dq,∗(Rq) the space of even functions infinitely q-differentiable on Rq with compact support in Rq. We equip this space with the topology of the uniform convergence of the functions and their q-derivatives. • Cq,∗,0(Rq) the space of even functions f defined on Rq continuous on 0, infinitely q-differentiable and lim x→∞ f(x) = 0, ‖f‖Cq,∗,0 = sup x∈Rq |f(x)| < +∞. (39) • Hq,∗(Rq) the space of even functions f defined on Rq continuous on 0 with compact support such that ‖f‖Hq,∗ = sup x∈Rq |f(x)| < +∞. (40) 2.5 q-Bessel function The following properties of the normalized q-Bessel function is given (see [13]) by jα(x;q 2 ) = Γq2(α + 1) ∞∑ k=0 (−1)kqk(k−1) Γq2(k + 1)Γq2(α + k + 1) ( x 1 + q ) 2k. (41) This function is bounded and for every x ∈ Rq and α > − 1 2 we have |jα(x;q 2 )| ≤ 1 (q;q2)2 ∞ , (42) ( 1 x Dq ) jα(.;q 2)(x) = − (1 − q) (1 − q2α+2) jα+1(qx;q 2), (43) ( 1 x Dq ) (x2αjα(x;q 2)) = 1 − q2α 1 − q x2(α−1)jα−1(x;q 2), (44) |Dqjα(x;q 2)| ≤ (1 − q) (1 − q2α+2) x (q;q2)2 ∞ . (45) We remark that for λ ∈ C, the function jα(λx;q 2) is the unique solution of the q-differential system    ∆q,αU(x,q) = −λ 2U(x,q), U(0,q) = 1; Dq,xU(x,q)|x=0 = 0, (46) 130 Belgacem Selmi CUBO 17, 2 (2015) where ∆q,α is the q-Bessel operator defined by ∆q,αf(x) = 1 x2α+1 Dq [ x2α+1Dqf ] (q−1x) (47) = q2α+1∆qf(x) + 1 − q2α+1 (1 − q)q−1x Dqf(q −1x), (48) where ∆qf(x) = Λ −1 q D 2 qf(x) = (D 2 qf)(q −1x), (49) and for k ∈ N and λ ∈ Rq,+, ∆kq,xjα(λx;q 2) = (−1)kλ2kjα(λx;q 2). (50) 2.6 q-Bessel Translation operator Tαq,x,x ∈ Rq,+ is the q-generalized translation operator associated with the q-Bessel transform is introduced in [13] and rectified in [17], where it is defined by the use of Jackson’s q-integral and the q-shifted factorial as Tαq,xf(y) = ∫+∞ 0 f(t)Dα,q(x,y,t)t 2α+1dqt, α > −1 (51) with Dα,q(x,y,z) = c 2 α,q ∫+∞ 0 jα(xt;q 2)jα(yt;q 2)jα(zt;q 2)t2α+1dqt where cα,q = 1 1 − q (q2α+2;q2)∞ (q2;q2)∞ . In particular the following product formula holds Tαq,xjα(y,q 2) = jα(x,q 2)jα(y,q 2). It is shown in [18] that for f ∈ L1α,q(Rq,+), T α q,xf ∈ L 1 α,q(Rq,+) and ||Tαq,xf||1,α,q = ||f||1,α,q. 2.7 The q-convolution and the q-Bessel Fourier transform The q-Bessel Fourier transform Fα,q and the q-Bessel convolution product are defined for suitable functions f,g as follows Fα,q(f)(λ) = ∫ ∞ 0 f(x)jα(λx;q 2 )dqσ(x), CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 131 f ∗α,q g(x) = ∫+∞ 0 Tαq,xf(y)g(y)dqσ(y). The q-Bessel Fourier transform Fα,q is a modified version of the q-analogue of the Hankel transform defined in [15]. It is shown in [13, 17, 14], that the q-Bessel Fourier transform Fα,q satisfies the following properties: Proposition 2.1. If f ∈ L1α,q(Rq,+), then Fα,q(f) ∈ Cq,∗,0(Rq,+) and ‖Fα,q(f)‖Cq,∗,0 ≤ Bα,q||f||1,α,q. where Bα,q = 1 (1 − q) (−q2;q2)∞(−q 2α+2;q2) (q2;q2)∞ . Proposition 2.2. Given two functions f, g ∈ L1α,q(Rq,+), then f ∗α,q g ∈ L 1 α,q(Rq,+), and Fα,q(f ∗α,q g) = Fα,q(f)Fα,q(g). Theorem 2.3. (Inversion formula) 1. If f ∈ L1α,q(Rq,+) such that Fα,q(f) ∈ L 1 α,q(Rq,+), then for all x ∈ Rq,+, we have f(x) = ∫ ∞ 0 Fα,q(f)(y)jα(xy;q 2 )dqσ(y). 2. Fα,q(f) is an isomorphism of S∗,q(Rq) and F 2 α,q(f) = Id. • Note that the inversion formula is valid for f ∈ L1α,q(Rq,+) without the additional condition Fα,q(f) ∈ L 1 α,q(Rq,+). Fα,q(f) can be extended to L 2 α,q(Rq,+) and we have the following theorem: Theorem 2.4. (q-Plancherel theorem ) Fα,q(f) is an isomorphism of L 2 α,q(Rq,+), we have ||Fα,q(f)||2,α,q = ||f||2,α,q, for f ∈ L 2 α,q(Rq,+) and F−1α,q(f) = Fα,q(f). Proposition 2.5. (i) For f ∈ L p α,q(Rq,+), p ∈ [1,∞[, g ∈ L 1 α,q(Rq,+) we have f ∗α,q g ∈ L p α,q(Rq,+) and ||f ∗αq g||p,α,q ≤ ||f||p,α,q||g||1,α,q. (ii) ∫ ∞ 0 Fα,q(f)(ξ)g(ξ)dqσ(ξ) = ∫ ∞ 0 f(ξ)Fα,q(g)(ξ)dqσ(ξ); f,g ∈ L 1 α,q(Rq,+). 132 Belgacem Selmi CUBO 17, 2 (2015) (iii) Fα,q(T α q,xf)(ξ) = jα(ξx;q 2)Fα,q(f)(ξ); f ∈ L 1 α,q(Rq,+). Specially, we choose q ∈ [0,q0] where q0 is the first zero of the function [17]: q 7→ 1φ1(0,q,q;q) under the condition log(1−q) log q ∈ Z. Definition 2.6. [11, 9] A bounded complex even measure µ on Rq is a bounded linear functional µ on Hq,∗(Rq), i.e., for all f in Hq,∗(Rq), we have |µ(f)| ≤ C‖f‖Hq,∗, (52) where C > 0 is a positive constant. Denote the space of all such measure by M ′ (Rq,+). Note that µ ∈ M ′ (Rq,+) can be identified with a function µ̃ on R̃q,+ such that µ̃ restricted to Rq,+ is L 1 α,q(Rq,+) : µ(f) = µ({0})f(0) + ∫ ∞ 0 µ̃(x)f(x)dq(x), (f ∈ Hq,∗(Rq)). For µ ∈ M ′ (Rq,+) denote ‖µ‖ = |µ|(Rq,+) where |µ| is the absolute value of µ. Definition 2.7. The q-Bessel Fourier transform of a measure µ in M ′ (Rq,+) is defined for all ϕ ∈ Sq,∗(Rq) by Fα,qµ(λ) = bα,q ∫+∞ 0 jα(λx;q 2)dµ(x). (53) The q-Bessel convolution product of a measure µ ∈ M ′ (Rq,+) and a suitable function f on Rq,+ is defined by µ ∗α,q f(x) = ∫ ∞ 0 Tαq,xf(y)dµ(y). (54) Proposition 2.8. (1) The q-Bessel Fourier transform Fα,q of a measure µ in M ′ (Rq,+) is the q-tempered distribution Fα,qµ given by: 〈Fα,qµ,ϕ〉 = 〈µ,Fα,qϕ〉 = ∫+∞ 0 Fα,qϕ(λ)dqµ(λ). (55) (2) For all x,λ ∈ Rq,+ we have Tαq,xFα,qµ(λ) = bα,q ∫+∞ 0 jα(xt;q 2)jα(λt;q 2)dqµ(t). (56) (3) For all µ ∈ M ′ (Rq,+), Fα,qµ is continuous on Rq,+, and lim λ→∞ Fα,qµ(λ) = µ({0}). (57) Fα,q maps one to one M ′ (Rq,+) into Cb(Rq,+), (the space of continuous and bounded func- tions on Rq,+). CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 133 (4) If µ ∈ M ′ (Rq,+) and f ∈ L p α,q(Rq,+), p = 1,2 then µ ∗α,q f ∈ L p α,q(Rq,+) and ‖µ ∗α,q f‖p,α,q ≤ ‖µ‖‖f‖p,α,q. (58) (5) For all µ ∈ M ′ (Rq,+) and f ∈ L p α,q(Rq,+), p = 1,2 we have Fα,q(µ ∗α,q f) = Fα,q(µ)Fα,q(f). (59) Definition 2.9. Let µ ∈ M ′ (Rq,+) and a > 0. We define the q-dilated measure µa of µ by ∫ ∞ 0 ϕ(x)dqµa(x) = ∫ ∞ 0 ϕ(ax)dqµ(x), ϕ ∈ Hq,∗(Rq). (60) Proposition 2.10. (i) When µ = f(x)x2α+1dqx, with f ∈ L 1 α,q(Rq,+), the measure µa, a > 0, is given by the function fa(x) = 1 a2α+2 f( x a ), x ≥ 0. (61) (ii) Let µ ∈ M ′ (Rq,+), then Fα,q(µa)(λ) = Fα,q(µ)(aλ), for all λ ≥ 0. (62) (iii) For µ ∈ M ′ (Rq,+) and f ∈ L p α,q(Rq,+),p = 1,2 we have lim a→0 µa ∗α,q f = µ(R̃q,+)f. (63) where the limit is in L p α,q(Rq,+). (iv) Let g ∈ L1α,q(Rq,+) and f ∈ L p α,q(Rq,+), 1 < p < ∞. Then lim a→∞ f ∗α,q ga = 0 (64) where the limit is in L p α,q(Rq,+). Proof. Statement of (i) and (ii) are obvious. A standard argument gives (iii). Let us verify (iv). If f,g ∈ Dq,∗(Rq) then by (58) and (61) we have ‖f ∗α,q ga‖p,α,q ≤ ‖f‖1,α,q‖ga‖p,α,q = a −2(α+1)(p−1) p ‖f‖1,α,q‖g‖p,α,q → 0, as a → ∞. For arbitrary g ∈ L1α,q(Rq,+) and f ∈ L p α,q(Rq,+) the result follows by density. ✷ Given a measure µ ∈ M ′ (Rq,+). Denote cµ,α,q = ∫ ∞ 0 Fα,q(µ)(λ) dqλ λ . (65) 134 Belgacem Selmi CUBO 17, 2 (2015) 3 q-Calderón’s formula for functions In this section, we establish the q-Calderón’s reproducing identity for functions using the proper- ties of q-Fourier Bessel transform Fα,q and q-Bessel convolution ∗α,q. Theorem 3.1. Let ϕ and ψ ∈ L1α,q(Rq,+) be such that following admissibility condition holds ∫ ∞ 0 Fα,q(ϕ)(ξ)Fα,q(ψ)(ξ) dqξ ξ = 1 (66) then for all f ∈ L1α,q(Rq,+), the following Calderón’s reproducing identity holds: f(x) = ∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t . (67) Proof. Taking q-Bessel Fourier transform of the right-hand side of (67), we get Fα,q [∫ ∞ 0 (f ∗α,q ϕt ∗α,q ψt)(x) dqt t ] (ξ) = ∫ ∞ 0 Fα,q(f)(ξ)Fα,q(ϕt)(ξ)Fα,q(ψt)(ξ) dqt t = Fα,q(f)(ξ) ∫ ∞ 0 Fα,q(ϕt)(ξ)Fα,q(ψt)(ξ) dqt t = Fα,q(f)(ξ) ∫ ∞ 0 Fα,q(ϕ)(tξ)Fα,q(ψ)(tξ) dqt t = Fα,q(f)(ξ). Now, by putting tξ = s, we get ∫ ∞ 0 Fα,q(ϕ)(tξ)Fα,q(ψ)(tξ) dqt t = ∫ ∞ 0 Fα,q(ϕ)(s)Fα,q(ψ)(s) dqs s = 1. Hence, the result follows. ✷ The equality (67) can be interpreted in the following L2-sense. Theorem 3.2. Suppose ϕ ∈ L1α,q(Rq,+) and satisfies ∫ ∞ 0 [Fα,q(ϕ)(ξ)] 2 dqξ ξ = 1. (68) For f ∈ L1α,q(Rq,+) ∩ L 2 α,q(Rq,+), suppose that fε,δ(x) = ∫δ ε (f ∗α,q ϕt ∗α,q ϕt)(x) dqt t (69) then ‖fε,δ − f‖2,α,q −→ 0 as ε → 0 and δ → ∞. (70) CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 135 Proof. Taking q-Bessel Fourier transform of both sides of (69) and using Fubini’s theorem, we get Fα,q(f ε,δ )(ξ) = Fα,q(f)(ξ) ∫δ ε [Fα,q(ϕ)(tξ)] 2 dqt t by Proposition 2.5, we have ‖ϕt ∗α,q ϕt ∗α,q f‖2,α,q ≤ ‖ϕt ∗α,q ϕt‖1,α,q‖f‖2,α,q ≤ ‖ϕt‖ 2 1,α,q‖f‖2,α,q. Now using above inequality, Minkowski’s inequality and relation (29), we get ‖fε,δ‖22,α,q = ∫ ∞ 0 | ∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t |2dqσ(x) ≤ ∫δ ε ∫ ∞ 0 |(ϕt ∗α,q ϕt ∗α,q f)(x)| 2dqσ(x) dqt t ≤ ∫δ ε ‖ϕt ∗α,q ϕt ∗α,q f‖2,α,q dqt t ≤ ‖ϕt‖ 2 1,α,q‖f‖2,α,q ∫δ ε dqt t = ‖ϕt‖ 2 1,α,q‖f‖2,α,q logq( δ ε ). Hence, by Theorem 2.4, we get lim ε → 0 δ → ∞ ‖fε,δ − f‖22,α,q = lim ε → 0 δ → ∞ ‖Fα,q(f ε,δ) − Fα,q(f)‖ 2 2,α,q = lim ε → 0 δ → ∞ ∫ ∞ 0 |Fα,q(f)(ξ) ( 1 − ∫δ ε [Fα,q(ϕ)(tξ)] 2 dqt t ) |2dqσ(x) = 0. Since |Fα,q(f)(ξ) ( 1 − ∫δ ε [Fα,q(ϕ)(tξ)] 2 dqt t ) | ≤ |Fα,q(f)(ξ)|, therefore, by the dominated con- vergence theorem, the result follows. ✷ The reproducing identity (67) holds in the pointwise sense under different sets of nice condi- tions. Theorem 3.3. Suppose f, Fα,qf ∈ L 1 α,q(Rq,+). Let ϕ ∈ L 1 α,q(Rq,+) and satisfies ∫ ∞ 0 [Fα,qϕ(tξ)] 2 dqt t = 1 (71) 136 Belgacem Selmi CUBO 17, 2 (2015) then lim ε → 0 δ → ∞ fε,δ(x) = f(x), (72) where fε,δ is given by (69). Proof. by Proposition 2.5, we have ‖ϕt ∗α,q ϕt ∗α,q f‖1,α,q ≤ ‖ϕt‖ 2 1,α,q‖f‖1,α,q. Now ‖fε,δ‖1,α,q = ∫ ∞ 0 | ∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t |dqσ(x) ≤ ∫δ ε ∫ ∞ 0 |(ϕt ∗α,q ϕt ∗α,q f)(x)|dqσ(x) dqt t ≤ ∫δ ε ‖ϕt ∗α,q ϕt ∗α,q f‖1,α,q dqt t ≤ ‖ϕt‖ 2 1,α,q‖f‖1,α,q logq( δ ε ). Therefore, fε,δ ∈ L1α,q(Rq,+). Also using Fubini’s theorem and taking q- Bessel Fourier transform of fε,δ, we get Fα,qf ε,δ(ξ) = ∫ ∞ 0 jα(xξ;q 2) (∫δ ε (ϕt ∗α,q ϕt ∗α,q f)(x) dqt t ) dqσ(x) = ∫δ ε ∫ ∞ 0 jα(xξ;q 2 )(ϕt ∗α,q ϕt ∗α,q f)(x)dqσ(x) dqt t = ∫δ ε Fα,qϕt(ξ)Fα,qϕt(ξ)Fα,qf(ξ) dqt t = Fα,qf(ξ) ∫δ ε [Fα,qϕ(tξ)] 2 dqt t . Therefore by (71), |Fα,qf ε,δ(ξ)| ≤ |Fα,qf(ξ)|. It follows that Fα,qf ε,δ ∈ L1α,q(Rq,+). By inversion, we have f(x) − fε,δ(x) = ∫ ∞ 0 jα(xξ;q 2 ) [ Fqf(ξ) − Fα,qf ε,δ (ξ) ] dqσ(ξ). (73) Putting gε,δ(x,ξ) = jα(xξ;q 2) [ Fα,qf(ξ) − Fα,qf ε,δ(ξ) ] (74) = jα(xξ;q 2)Fqf(ξ) [ 1 − ∫δ ε [Fα,qϕ(tξ)] 2 dqt t ] , CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 137 we get f(x) − fε,δ(x) = ∫ ∞ 0 jα(xξ;q 2) [ Fα,qf(ξ) − Fα,qf ε,δ(ξ) ] dqσ(ξ) = ∫ ∞ 0 gε,δ(x,ξ)dqσ(ξ). Now using (71) and (74), we get lim ε → 0 δ → ∞ gε,δ(x,ξ) = 0. (75) Since |gε,δ(x,ξ)| ≤ 1 (q;q2)2 ∞ |Fα,qf(ξ)|, the dominated convergence theorem yields the result. ✷ 4 q-Calderón’s formula for finite measures It is now possible to define analogues to (2) for the q-Bessel convolution ∗α,q and investigate its convergence in the L2α,q(Rq,+) q-norm. To this end we need some technical lemmas Lemma 4.1. Let µ ∈ M ′ (Rq,+), for 0 < ε < δ < ∞ define Gε,δ(x;q 2) = µ([x δ , x ε ]) x2α+2 , x > 0 (76) and Kε,δ(λ;q 2) = ∫δ ε Fα,q(µ)(qaλ) dqa a , λ ≥ 0. (77) Then Gε,δ ∈ L 1 α,q(Rq,+) and Fα,q(Gε,δ)(λ;q 2 ) = Kε,δ(λ;q 2 ) − µ({0}) logq( δ ε ), (78) where logq is given by (25). Proof. We have by (25) and (29), | ∫ ∞ 0 Gε,δ(x;q 2 )x2α+1dqx| ≤ ∫ ∞ 0 ( ∫ x ε x δ dq|µ|(y)) dqx x = ∫ ∞ 0 [ ∫ x ε 0 dq|µ|(y) − ∫ x δ 0 dq|µ|(y)] dqx x = ∫ ∞ 0 [ ∫ ∞ qεy dqx x − ∫ ∞ qδy dqx x ]dq|µ|(y) = ∫ ∞ 0 logq( ε δ )dq|µ|(y) = |µ|(R̃q,+) logq( ε δ ) < ∞. 138 Belgacem Selmi CUBO 17, 2 (2015) Using again relation (29) and q-Fubini’s theorem we obtain Fα,q(Gε,δ)(λ) = ∫ ∞ 0 ∫ x ε x δ dqµ(y)jα(λx;q 2) dqx x = ∫ ∞ 0 ∫qδy qεy jα(λx;q 2) dqx x dqµ(y) = ∫ ∞ 0 ∫qδ qε jα(λxy;q 2 ) dqx x dqµ(y) = ∫qδ qε ∫ ∞ 0 jα(λxy;q 2)dqµ(y) dqx x = ∫qδ qε Fα,qµ(λx) − µ({0}) dqx x = ∫δ ε Fα,qµ(qλx) − µ({0}) dqx x = Kε,δ(λ;q 2 ) − µ({0}) logq( δ ε ). ✷ Lemma 4.2. Let µ ∈ M ′ (Rq,+), then for f ∈ L p α,q(Rq,+),p = 1,2 and 0 < ε < δ < ∞, the function fε,δ(x;q2) = ∫δ ε f ∗α,q µa(x;q 2) dqa a (79) belongs to L p α,q(Rq,+) and has the form fε,δ(x;q2) = f ∗α,q Gqε,qδ(x;q 2) + µ({0})f(x) logq( δ ε ). (80) where Gε,δ is given by (4.1). Proof. Applying q-Fubini’s theorem we get fε,δ(x) = ∫δ ε ∫ ∞ 0 Tαq,xf(ay)dqµ(y) dqa a = ∫ ∞ 0 ∫δ ε Tαq,ayf(x) dqa a dqµ(y) = f(x)µ({0}) logq( δ ε ) + ∫ R̃q,+ ∫δy εy Tαq,xf(a) dqa a dqµ(y) = f(x)µ({0}) logq( δ ε ) + ∫ R̃q,+ Tαq,xf(a)( ∫q a ε q a δ dqa a )dqµ(y) = f(x)µ({0}) logq( δ ε ) + f ∗α,q Gqε,qδ(x). CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 139 From this relation, inequality (58) and Lemma 4.1 we deduce that fε,δ belongs to L p α,q(Rq,+). ✷ Lemma 4.3. Let µ ∈ M ′ (Rq,+), then for f ∈ L 2 α,q(Rq,+), we have Fα,q(f ε,δ)(λ;q2) = Fq(f)(λ;q 2)Kqε,qδ(λ;q 2), (81) where Kε,δ, is the function defined in (67). Proof. This follows from (59), (67) and (80). ✷ Theorem 4.4. Let µ ∈ M ′ (Rq,+), be such that the q-integral cµ,α,q = ∫ ∞ 0 Fα,q(µ)(λ) dqλ λ (82) be finite. Then for all f ∈ L2α,q(Rq,+), we have lim ε → 0 δ → ∞ ‖fε,δ − cµ,α,qf‖2,α,q = 0. (83) Proof. By identity (81) and Theorem 2.4 we have ‖fε,δ − cµ,α,qf‖ 2 2,α,q = ‖Fα,q(f ε,δ ) − cµ,α,qFα,q(f)‖ 2 2,α,q = ‖Fα,q(f)[Kε,δ − cµ,α,q]‖ 2 2,α,q. Or lim ε → 0 δ → ∞ Kε,δ(λ) = cµ,α,q, for all λ > 0 the result follows from the dominate convergence theo- rem. ✷ Lemma 4.5. Let µ ∈ M ′ (Rq,+), be such that the q-integral ∫ ∞ 0 |µ([0,y])| dqy y (84) be finite. Then the q-integral cµ,α,q is finite and admits the representation cµ,α,q = ∫ ∞ 0 µ([0,y]) dqy y . (85) Proof. From (76) we have Gε,δ = µ([x δ , x ε ]) x2α+2 = Gε − Gδ, (86) 140 Belgacem Selmi CUBO 17, 2 (2015) where G(y) = µ([0,y]) y2α+2 (87) and Gε, Gδ the dilated function of G. Since G ∈ L 1 α,q(Rq,+), we deduce from (62) and (78) that Fα,qGε,δ(λ) = ∫δλ ελ Fα,qµ(a) dqa a − µ({0}) logq( δ ε ) (88) = Fα,qG(ελ) − Fα,qG(δλ), for all λ > 0. Or (84) implies necessarily µ({0}) = 0. Hence when ε = 1 and δ → ∞, a combination of (88) and (57) gives Fα,qG(λ) = ∫ ∞ λ Fα,qµ(a) dqa a , for all λ > 0. (89) Now the result follows from Formula (84) by using the continuity of Fα,q(µ). ✷ Theorem 4.6. Let µ ∈ M ′ (Rq,+) such that ∫ ∞ 0 |µ([0,y])| dqy y (90) is finite and f ∈ L2α,q(Rq,+). Then lim ε → 0 δ → ∞ ‖fε,δ − cµ,α,qf‖2,α,q = 0. (91) Proof. By (80) and (86) we have fε,δ = f ∗α,q Gε − f ∗α,q Gδ, (92) where G is as in (87). Equation (91) is now a consequence of Proposition 2.5. ✷ Received: July 2014. Accepted: May 2015. References [1] A. P. Calderón, Intermediate spaces and interpolation, the complex method,Studia Math, 24 (1964), 113-190. [2] B. Rubin and E. Shamir, Calderón’s reproducing formula and singular integral operators on areal line,Integral Equations Operators Theory, 21 (1995), 77-92. [3] B. Rubin, Fractional Integrals and Potentials ,Logman, Harlow, (1996). CUBO 17, 2 (2015) Calderón’s reproducing Formula For q-Bessel operator 141 [4] M.A. Mourou and K. Trimèche, Calderón’s reproducing formula associated with the Bessel Op- erator ,J. Math.Anal. Appl., 219 (1998), 97-109. [5] P.S. Pathak and G. Pandey, Calderón’s reproducing formula for Hankel convolution,Inter. J. Math. and Math. Sci., vol 2006 issue 5, Article ID 24217. P.1-7. [6] J. L. Ansorena and O. Blasco, Characterization of weighted Besov spaces, Math. Nachr, 171 (1995), 5-17. [7] G. Gasper and M. Rahman, Basic hypergeometric series, 2nd edn. Cambridge University Press, 2004. [8] H. Q. Bui, M. Paluszynski and M. H. Taibleson , A maximal function charaterization of weighted Besov spaces-Lipschitz and Triebel-Lizorkin spaces, Studia. Math, 119 (1996), 219-246. [9] A. Nemri and B. Selmi, Calderón type formula in Quantum Calculus, Indagationes Mathematicae, vol 24, issues 3, (2013), 491-504. [10] A. Nemri and B. Selmi, Some weighted Besov Spaces in Quantum Calculus, submitted. [11] Nemri A, Distribution of positive type in Quantum Calculus, J. Non Linear. Math. Phys., 4(2006)566- 583. [12] A. Fitouhi and A. Nemri, Distribution And Convolution Product in Quantum Calculus, Afri. Di- aspora J. Math, 7(2008),nï¿ 1 2 1, 39-57. [13] A. Fitouhi, M. Hamza and F. Bouzeffour, The q-jα Bessel function, J. Approx. Theory 115 (2002), 114-116. [14] M.Haddad, Hankel transform in Quantum Calculus and applications, fractional Calculus and Applied Analysis, Vol.9,issues 4 (2006), 371-384. [15] T. H. Koornwinder and R. F. Swarttouw, On q-Analogues of the Fourier and Hankel transforms , Trans. Amer Math. Soc. 333, (1992), 445-461. [16] T. H. Koornwinder, q-Special functions, a tutorial arXiv:math/9403216v1. [17] A. Fitouhi, L. Dhaouadi and J. El Kamel, Inequalities in q-Fourier analysis , J. Inequal. Pure Appl. Math, 171 (2006), 1-14. [18] A. Fitouhi and L. Dhaouadi, Positivity of the Generalized Translation Associated to the q-Hankel Transform, Constr. Approx, 34 (2011), 453-472. [19] V.G. Kac and P. Cheeung, Quantum calculus, Universitext, Springer-Verlag, New York, (2002). Introduction Preliminaries q-Exponential series q-Derivative and q-Integral The q-gamma function Some q-functional spaces q-Bessel function q-Bessel Translation operator The q-convolution and the q-Bessel Fourier transform q-Calderón's formula for functions q-Calderón's formula for finite measures