Weierstrass Transform Associated with the Hankel Operator ∗

Bulletin of Mathematical analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 1-16

Weierstrass transform associated with the Hankel operator ∗

Slim Omri, & Lakhdar Tannech Rachdi

Abstract Using reproducing kernels for Hilbert spaces, we give best approximation for the Weierstrass transform associated with the Hankel transform. Also, estimates of extremal functions are checked.

1 Introduction

The Hankel transform Hµ, µ > −1/2, is defined for all integrable functions on r2µ+1 [0, +∞[ with respect to the measure dr as 2µΓ(µ + 1) Z +∞ 1 2µ+1 Hµ(λ) = µ f(r)jµ(λr)r dr, 2 Γ(µ + 1) 0 where jµ is the modified Bessel function of the first kind and index µ.

Many harmonic analysis results related to the transform Hµ are established in [6, 9, 13, 18, 20, 21].

Our purpose in this work is to deﬁne and study the Weierstrass transform Wµ,t associated with the Hankel transform Hµ.

This transform is deﬁned by Z +∞ s2µ+1 Wµ,t(f)(r) = Et(r, s)f(s) µ ds, 0 2 Γ(µ + 1) where Et(r, s), t > 0 is the heat kernel associated with the Hankel transform which will be deﬁned later. This integral transform which generalizes the usual Weierstrass transform [11, 15, 16], solves the heat problem

∂u `µ(u)(r, t) = ∂t (r, t), u(r, 0) = f(r),

∗Mathematics Subject Classifications: 44A15, 46E22, 35K05. Key words: Weierstrass transform, best approximation, Hankel transform, reproducing kernel, extremal function. c 2009 Universiteti i Prishtinës,Prishtinë,Kosovë. Submitted March, 2009. Published May, 2009.

1 2 Weierstrass transform associated with the Hankel operator

where `µ is the singular diﬀerential operator deﬁned on ]0, +∞[ by ∂2 2µ + 1 ∂ ` = + . µ ∂r2 r ∂r Building on the ideas of Saitoh, Matsuura, Fujiwara and Yamada [5, 12, 14, 15, 16], and using the theory of reproducing kernels [2], we give a best approximation of this transform and nice estimates of the associated extremal function.

r2µ+1 Let L2([0, +∞[, dr) be the Hilbert space of square integrable func- 2µΓ(µ + 1) r2µ+1 tions on [0, +∞[ with respect to the measure dr, and h.|.i its inner 2µΓ(µ + 1) µ product.

µ For ν ∈ R, we consider the Sobolev type space Hν ([0, +∞[), consisting of func- r2µ+1 tions f ∈ L2([0, +∞[, dr) such that the function 2µΓ(µ + 1)

2 ν/2 λ 7−→ (1 + λ ) Hµ(f)(λ), r2µ+1 belongs to the space L2([0, +∞[, dr). Then for ν > µ+1, Hµ([0, +∞[) 2µΓ(µ + 1) ν is the Hilbert space when equipped with the inner product

Z +∞ 2µ+1 2 ν λ hf|giν = (1 + λ ) Hµ(f)(λ)Hµ(g)(λ) µ dλ. 0 2 Γ(µ + 1) Moreover, the kernel

Z +∞ 2µ+1 jµ(λr)jµ(λs) λ Kν (r, s) = 2 ν µ dλ, 0 (1 + λ ) 2 Γ(µ + 1) µ is a reproducing kernel of the space Hν ([0, +∞[), where

+∞ 2µΓ(µ + 1) X (−1)n z j (z) = J (z) = Γ(µ + 1) ( )2n, z ∈ , µ zµ µ n!Γ(µ + n + 1) 2 C n=0 and Jµ is the Bessel function of the ﬁrst kind and index µ. Using the properties of the Hankel transform Hµ and its connection with the convolution product, we show that the Weierstrass transform Wµ,t is a bounded linear operator from r2µ+1 Hµ([0, +∞[) into L2([0, +∞[, dr) and that for all f ∈ Hµ([0, +∞[), ν 2µΓ(µ + 1) ν

kWµ,t(f)k2,µ 6 ||f||ν . µ Next, for ξ > 0, we deﬁne on the space Hν ([0, +∞[), the new inner product by setting hf|giν,ξ = ξhf|giν + hWµ,t(f)|Wµ,t(g)iµ. S. Omri, & L. T. Rachdi 3

µ We show that Hν ([0, +∞[) equipped with the inner product h.|.iν,ξ, is a Hilbert space and we exhibit a reproducing kernel, that is Z +∞ j (λr)j (λs) λ2µ+1 K (r, s) = µ µ dλ. ν,ξ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e 2 Γ(µ + 1) The last section of this paper is devoted to study the extremal function. More r2µ+1 precisely, for all ν > µ + 1, ξ > 0 and g ∈ L2([0, +∞[, dr), the 2µΓ(µ + 1) inﬁmum of

n 2 2 µ o ξ||f||ν + ||g − Wµ,t(f)||2,µ; f ∈ Hν ([0, +∞[) ,

∗ is attained at one function fξ,g, called the extremal function. We establish also, the following estimates

µ • For all f ∈ Hν ([0, +∞[) and g = Wµ,t(f), ∗ lim fξ,g − f = 0. ξ→0+ ν

µ • For all f ∈ Hν ([0, +∞[) and g = Wµ,t(f), ∗ lim fξ,g(r) = f(r), uniformly. ξ→0+

2 The Hankel transform

We denote by

• dγµ the measure deﬁned on [0, +∞[ by r2µ+1 dγ (r) = dr, µ 2µΓ(µ + 1)

p • L (dγµ) , p ∈ [1, +∞], the space of measurable functions f on [0, +∞[ satisfying

1 Z +∞ p p kfkp,µ = |f(r)| dγµ(r) < +∞, if p ∈ [1, +∞[; 0 kfk∞,µ = ess sup |f(r)| < +∞, if p = +∞. r∈[0,+∞[

2 • h.|.iµ the inner product on L (dγµ) deﬁned by Z +∞ hf|giµ = f(r)g(r)dγµ(r). 0

•C∗,0(R) the space of even continuous functions f on R such that lim f(r) = 0. |r|→+∞ 4 Weierstrass transform associated with the Hankel operator

Let `µ be the Bessel operator deﬁned on ]0, +∞[ by 2µ + 1 ` (u) = u00 + u0, µ r then for all λ ∈ C, the following problem,

 2  `µ(u) = −λ u, u(0) = 1,  u0(0) = 0,

admits a unique solution given by jµ(λ.), where

+∞ 2µΓ(µ + 1) X (−1)n z j (z) = J (z) = Γ(µ + 1) ( )2n, z ∈ , (2.1) µ zµ µ n!Γ(µ + n + 1) 2 C n=0

and Jµ is the Bessel function of the ﬁrst kind and index µ [1, 3, 10, 22].

The eigenfunction jµ satisﬁes the following properties

• The function jµ has the Mehler integral representation, for all x ∈ R

 2Γ(µ + 1) Z 1  √ (1 − t2)µ−1/2 cos x t dt, if µ > −1/2; jµ(x) = πΓ(µ + (1/2)) 0  cos x, if µ = −1/2.

• For all n ∈ N and x ∈ R

(n) jµ (x)| 6 1. (2.2)

• The function jµ satisﬁes the product formula [10, 22], for all r, s ∈ [0, +∞[  Z π Γ(µ + 1) p 2 2 2µ  jµ( r + s + 2rs cos θ)(sin θ) dθ, if µ > −1/2; j (r)j (s) = Γ(µ + 1/2)Γ(1/2) 0 µ µ j (r + s) + j (r − s)  −1/2 −1/2 , if µ = −1/2. 2 Using the product formula, we will deﬁne and study the Hankel translation operator and the convolution product.

µ Deﬁnition 2.1 1) For all r ∈ [0, +∞[, the Hankel translation operator τr is p deﬁned on L (dγµ) by  Z π Γ(µ + 1) p 2 2 2µ  f( r + s + 2rs cos θ)(sin θ) dθ, if µ > −1/2; τ µ(f)(s) = Γ(µ + 1/2)Γ(1/2) 0 r f(r + s) + f(|r − s|)  , if µ = −1/2. 2 S. Omri, & L. T. Rachdi 5

1 2) The convolution product of f, g ∈ L (dγµ) is deﬁned by Z +∞ µ f ∗µ g(r) = τr (f)(s)g(s)dγµ(s). 0 The following assumptions hold

• The product formula can be written

µ ∀(r, s) ∈ [0, +∞[×[0, +∞[, τr (jµ(λ.))(s) = jµ(λr)jµ(λs). p µ • For all f ∈ L (dγµ), p ∈ [1, +∞], and for all r ∈ [0, +∞[ the function τr (f) p belongs to the space L (dγµ) and

µ kτr (f)kp,µ 6 kfkp,µ. (2.3) p • Let p, q, r ∈ [1, +∞] be such that 1/p+1/q = 1+1/r. For all f ∈ L (dγµ) and q r g ∈ L (dγµ), the function f ∗µ g belongs to L (dγµ) and we have the following Young inequality kf ∗µ gkr,µ 6 kfkp,µkgkq,µ. (2.4) 1 µ 1 • For all f ∈ L (dγµ) and λ ∈ [0, +∞[, the function τλ (f) belongs to L (dγµ) and we have Z +∞ Z +∞ µ τλ (f)(r)dγµ(r) = f(r)dγµ(r). (2.5) 0 0

1 Definition 2.2 The Hankel transform Hµ is defined on L (dγµ) by [17] Z +∞ Hµ(f)(λ) = f(r)jµ(rλ)dγµ(r), λ ∈ R, 0 where jµ is the modified Bessel function defined by the relation (2.1).

The Hankel transform satisﬁes the following properties

1 • For all f ∈ L (dγµ) the function Hµ(f) belongs to the space C∗,0(R) and

kHµ(f)k∞,µ 6 kfk1,µ. 1 • For all f ∈ L (dγµ) and r ∈ [0, ∞[ µ Hµ τr (f) (λ) = jµ(rλ)Hµ(f)(λ). (2.6) 1 • For f, g ∈ L (dγµ) Hµ(f ∗µ g) = Hµ(f)Hµ(g).

1 Theorem 2.3 (Inversion formula for Hµ) Let f ∈ L (dγµ) such that Hµ(f) ∈ 1 L (dγµ), then for almost every r ∈ [0, +∞[, we have Z +∞ f(r) = Hµ(f)(λ)jµ(λr)dγµ(λ). 0 6 Weierstrass transform associated with the Hankel operator

Theorem 2.4 (Plancherel theorem) The Hankel transform Hµ can be extended 2 to an isometric isomorphism from L (dγµ) onto itself. In particular for all 2 f, g ∈ L (dγµ), we have (Parseval equality) Z +∞ Z +∞ f(r)g(r)dγµ(r) = Hµ(f)(λ)Hµ(g)(λ)dγµ(λ). 0 0 1 2 Remark 2.5 i) Let f ∈ L (dγµ) and g ∈ L (dγµ), by the relation (2.4), the 2 function f ∗µ g belongs to L (dγµ), moreover

Hµ(f ∗µ g) = Hµ(f)Hµ(g). 2 ii) For all f, g ∈ L (dγµ) the function f ∗µ g belongs to the space C∗,0(R) and we have f ∗µ g = Hµ Hµ(f)Hµ(g) . (2.7)

3 Weierstrass transform associated with the Han- kel operator

In this section, we will deﬁne and study the Weierstrass transform associated with Hµ. For this we deﬁne some Hilbert spaces and we exhibit their reproducing kernels.

Let ν be a real number, ν > µ + 1. We denote by

µ 2 • Hν ([0, +∞[) the subspace of L (dγµ) formed by the functions f, such that the maps 2 ν/2 λ 7−→ (1 + λ ) Hµ(f)(λ), 2 belongs to L (dγµ).

µ • h.|.iν the inner product on Hν ([0, +∞[) deﬁned by Z +∞ 2 ν hf|giν = (1 + λ ) Hµ(f)(λ)Hµ(g)(λ)dγµ(λ). 0 µ • ||.||ν the norm of Hν ([0, +∞[) deﬁned by p ||f||ν = hf|fiν . Remark 3.1 For ν > µ + 1, the function 1 λ 7−→ , (1 + λ2)ν/2 2 µ belongs to L (dγµ). Hence for all f ∈ Hν ([0, +∞[), the function Hµ(f) belongs 1 to L (dγµ), then by inversion formula 2.3, we have for almost every r ∈ [0, +∞[ Z +∞ f(r) = Hµ(λ)jµ(λr)dγµ(λ). 0 S. Omri, & L. T. Rachdi 7

Proposition 3.2 For ν > µ + 1 the function Kν deﬁned on [0, +∞[×[0, +∞[ by Z +∞ jµ(λr)jµ(λs) Kν (r, s) = 2 ν dγµ(λ), 0 (1 + λ ) µ is a reproducing kernel of the space Hν ([0, +∞[), that is i) For all s ∈ [0, +∞[, the function

r 7−→ Kν (r, s),

µ belongs to Hν ([0, +∞[). µ ii) (The reproducing property) For all f ∈ Hν ([0, +∞[) and s ∈ [0, +∞[,

hf|Kν (., s)iν = f(s).

Proof. i) From Remark 3.1 and the relation (2.2), we deduce that for all s ∈ [0, +∞[, the function

j (λs) λ 7−→ µ , (1 + λ2)ν

1 2 belongs to L (dγµ) ∩ L (dγµ). Then, the function Kν is well deﬁned and by Theorem 2.3, we have j (λs) K (r, s) = H µ (r). ν µ (1 + λ2)ν

By Plancherel theorem, it follows that for all s ∈ [0, +∞[, the function Kν (., s) 2 belongs to L (dγµ), and we have j (λs) H K (., s) (λ) = µ . (3.8) µ ν (1 + λ2)ν

Again, by the relation (2.2) and Remark 3.1, it follows that the function

2 ν/2 λ 7−→ (1 + λ ) Hµ Kν (., s) (λ),

2 belongs to L (dγµ).

µ ii) Let f be in Hν ([0, +∞[). For every s ∈ [0, +∞[, we have Z +∞ 2 ν hf|Kν (., s)iν = (1+λ ) Hµ(f)(λ)Hµ(Kν (., s))(λ)dγµ(λ), 0 and by the relation (3.8), we get

Z +∞ hf|Kν (., s)iν = Hµ(f)(λ)jµ(λs)dγµ(λ). 0 The result follows from Remark 3.1. 8 Weierstrass transform associated with the Hankel operator

The heat equation associated with the Hankel transform is given by

∂ ` u(r, t) = u(r, t), (3.9) µ ∂t where `µ is the Bessel operator deﬁned above.

Let E be the kernel deﬁned by

Z +∞ −tλ2 E(r, t) = e jµ(rλ)dγµ(λ) (3.10) 0 2 e−r /4t = . (2t)µ+1

Then, the kernel E solves the equation (3.9).

Deﬁnition 3.3 The heat kernel associated with the Hankel transform is deﬁned by

µ Et(r, s) = τr E(., t) (s) (3.11)

2 2 e−(r +s )/4t irs = j ( ). (2t)µ+1 µ 2t

Then, we have the following properties

i) For all t > 0, Et > 0. ii) From the relations (2.3),(2.6),(3.10) and (3.11), for all t > 0, r ∈ [0, +∞[, 1 the function Et(r, .) belongs to L (dγµ) and for all λ ∈ [0, +∞[, we have

−tλ2 Hµ Et(r, .) (λ) = e jµ(λr).

iii) From the relations (2.3), (2.5), (3.10) and (3.11), for all t > 0 and 1 s ∈ [0, +∞[, the function Et(., s) belongs to L (dγµ) and we have

Z +∞ Z +∞ Et(r, s)dγµ(r) = E(r, t)dγµ(r) = 1. 0 0 iv) For all s ∈ [0, +∞[, the function

(r, t) 7−→ Et(r, s), solves the heat equation (3.9).

In the following, we shall deﬁne the Weierstrass transform associated with the Hankel transform and we establish some properties that we use later. S. Omri, & L. T. Rachdi 9

Deﬁnition 3.4 The Weierstrass transform associated with the Hankel trans- 2 form is deﬁned on L (dγµ), by Wµ,t(f)(r) = E(., t) ∗µ f (r) (3.12) Z +∞ = Et(r, s)f(s)dγµ(s). 0

For the classical Weierstrass transform, one can see [11, 15, 16].

2 Proposition 3.5 i) For all f ∈ L (dγµ), the function Wµ,t(f) solves the heat equation (3.9), with the initial condition

2 lim Wµ,t(f) = f, in L (dγµ). t→0+ ii) For all t > 0 and ν > µ + 1, the transform Wµ,t is a bounded linear operator µ 2 µ from Hν ([0, +∞[) into L (dγµ) and for all f ∈ Hν ([0, +∞[) we have

kWµ,t(f)k2,µ 6 ||f||ν .

Proof. i) From the relations (3.11), (3.12), the derivative’s theorem and the fact that for all s ∈ [0, +∞[, the function (r, t) 7−→ Et(r, s) solves the heat equation (3.9), we deduce that the function Wµ,t(f) is a solution of (3.9). The family E(., t) t>0 is an approximate identity, in particular for all f ∈ 2 L (dγµ) 2 lim E(., t) ∗µ f = f in L (dγµ). t→0

2 ii) From the relations (2.4) and (3.12), for all f ∈ L (dγµ), we have

kWµ,t(f)k2,µ = kE(., t) ∗µ fk2,µ

6 kE(., t)k1,µ kfk2,µ

= kfk2,µ = kHµ(f)k2,µ 6 kfkν .

Notations. For all positive real numbers ξ, t and for ν > µ + 1, we denote by µ • h.|.iν,ξ, the inner product deﬁned on the space Hν ([0, +∞[) by

hf|giν,ξ = ξhf|giν + hWµ,t(f)|Wµ,t(g)iµ.

µ µ • Hν,ξ([0, +∞[), the space Hν ([0, +∞[) equipped with the inner product h.|.iν,ξ and the norm

2 2 2 ||f||ν,ξ = ξ||f||ν + ||Wµ,t(f)||2,µ.

Then, we have the following main result [11, 15]. 10 Weierstrass transform associated with the Hankel operator

µ Theorem 3.6 For all ξ, t > 0 and ν > µ + 1, the Hilbert space Hν,ξ([0, +∞[) admits the following reproducing kernel, Z +∞ j (rλ)j (sλ) K (r, s) = µ µ dγ (λ), ν,ξ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e that is µ i) For all s ∈ [0, +∞[, the function r 7−→ Kν,ξ(r, s) belongs to Hν,ξ([0, +∞[). µ ii) (The reproducing property.) For all f ∈ Hν,ξ([0, +∞[) and s ∈ [0, +∞[,

hf|Kν,ξ(., s)iν,ξ = f(s). Proof. i) Let s ∈ [0, +∞[. From the inequality (2.2), we have

jµ(λs) 1 . ξ(1 + λ2)ν + e−2tλ2 6 ξ(1 + λ2)ν Then by the hypothesis ν > µ + 1, we deduce that for all s ∈ [0, +∞[, the function j (λs) λ 7−→ µ , ξ(1 + λ2)ν + e−2tλ2 1 2 belongs to L (dγµ) ∩ L (dγµ), and by the Plancherel theorem, the function j (λs) r 7−→ K (r, s) = H µ (r) (3.13) ν,ξ µ ξ(1 + λ2)ν + e−2tλ2

2 belongs to L (dγµ), moreover the function (1 + λ2)ν/2j (λs) λ 7−→(1 + λ2)ν/2H K (., s) (λ) = µ , µ ν,ξ ξ(1 + λ2)ν + e−2tλ2

2 belongs to L (dγµ).

This proves that for all s ∈ [0, +∞[, the function Kν,ξ(., s) belongs to the µ space Hν,ξ([0, +∞[). µ ii) Let f be in Hν,ξ([0, +∞[). By the relation (3.13), we get ! Z +∞ j (λs) hf|K (., s)i = (1 + λ2)ν H (f)(λ) × µ dγ (λ). ν,ξ ν µ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e (3.14) On the other hand, we have Wµ,t Kν,ξ(., s) (r) = E(., t) ∗µ Kν,ξ(., s) (r), and by the relations (2.7), (3.10) and (3.13), we get

2 j (λs)e−tλ W K (., s) (r) = H µ (r). (3.15) µ,t ν,ξ µ ξ(1 + λ2)ν + e−2tλ2 S. Omri, & L. T. Rachdi 11

By the same way

−tλ2 Wµ,t(f)(r) = Hµ e Hµ(f) (r). (3.16)

Thus,

−tλ2 2 j (λs)e hW (f)|W K (., s) i = hH e−tλ H (f) |H µ i . µ,t µ,t ν,ξ µ µ µ µ ξ(1 + λ2)ν + e−2tλ2 µ

Using the Parseval formula, we get

−tλ2 2 j (λs)e hW (f)|W K (., s) i = he−tλ H (f)| µ i . (3.17) µ,t µ,t ν,ξ µ µ ξ(1 + λ2)ν + e−2tλ2 µ

Combining the relations (3.14) and (3.17), we obtain

Z +∞ hf|Kν,ξ(., s)iν,ξ = Hµ(f)(λ)jµ(λs)dγµ(λ). 0 The desired result arises from Remark 3.1.

4 The Extremal Function

This section contains the main result of this paper, that is the existence and unicity of the extremal function related to the generalized Weierstrass transform studied in the previous section.

2 Theorem 4.1 Let ν > µ + 1, ξ > 0 and g ∈ L (dγµ). Then there is a unique ∗ µ function fξ,g ∈ Hν ([0, +∞[), where the inﬁmum of

n 2 2 µ o ξ||f||ν + ||g − Wµ,t(f)||2,µ, f ∈ Hν ([0, +∞[) ,

∗ is attained. Moreover, the extremal function fξ,g is given by

Z +∞ ∗ fξ,g(r) = g(s)Qξ(r, s)dγµ(s), (4.18) 0 where, 2 Z +∞ e−tλ j (λr)j (λs) Q (r, s) = µ µ dγ (λ). (4.19) ξ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e ∗ Proof. The existence and unicity of the extremal function fξ,g is given by [11, 15, 16]. On the other hand, we have

∗ fξ,g(s) = hg|Wµ,t Kν,ξ(., s) iµ, 12 Weierstrass transform associated with the Hankel operator and by (3.15), we obtain

2 j (λs)e−tλ f ∗ (s) = hg|H µ i ξ,g µ ξ(1 + λ2)ν + e−2tλ2 µ 2 Z +∞ Z +∞ e−tλ j (λr)j (λs) = g(r) µ µ dγ (λ) dγ (r) 2 ν −2tλ2 µ µ 0 0 ξ(1 + λ ) + e Z +∞ = g(r)Qξ(r, s)dγµ(r). 0 2 Corollary 4.2 Let ν > µ + 1, ξ > 0 and g ∈ L (dγµ). The extremal function ∗ fξ,g, satisﬁes the following inequality +∞ 2 Γ(ν − µ − 1) Z 2 f ∗ er |g(r)|2dγ (r). ξ,g 2,µ 6 2µ+4 µ ξ2 Γ(ν) 0 Proof. We have +∞ Z r2 r2 ∗ − 2 2 fξ,g(s) = e e g(r)Qξ(r, s)dγµ(r). 0 By H¨older’sinequality we get +∞ +∞ Z 2 Z 2 2 ∗ 2 −r r 2 |fξ,g(s)| 6 e dγµ(r) e |g(r)| Qξ(r, s) dγµ(r) . 0 0

Integrating over [0, +∞[ with respect to the measure dγµ(s), we obtain +∞ +∞ 2 Z 2 Z 2 f ∗ e−r dγ (r) er |g(r)|2 kQ (r, .)k2 dγ (r) . ξ,g 2,µ 6 µ ξ 2,µ µ 0 0 However, by the relation (4.19)

2 e−tλ j (λr) Q (r, s) = H µ (s), (4.20) ξ µ ξ(1 + λ2)ν + e−2tλ2 then by the Plancherel theorem Z +∞ −2tλ2 2 2 e |jµ(λr)| kQξ(r, .)k = dγµ(λ). 2,µ 2 2 0 ξ(1 + λ2)ν + e−2tλ 2 2 Since, a + b > 2ab, a, b > 0, and in virtue of the relation (2.2), it follows that Z +∞ 2 1 dγµ(λ) kQξ(r, .)k2,µ 6 2 ν . (4.21) 4ξ 0 (1 + λ ) We complete the proof by using the relations (4.20) and (4.21), and the fact that Z +∞ −r2 1 e dγµ(r) = µ+1 , 0 2 and Z +∞ dγµ(λ) Γ(ν − µ − 1) 2 ν = µ+1 . 0 (1 + λ ) 2 Γ(ν) S. Omri, & L. T. Rachdi 13

2 Corollary 4.3 Let ν > µ + 1. For all g1, g2 ∈ L (dγµ), we have

kg − g k ∗ ∗ 1 2 2,µ f − f √ . ξ,g1 ξ,g2 ν 6 2 ξ

Proof. Let ν > µ + 1. For all r ∈ [0, +∞[, the function

2 e−tλ λ 7−→ j (λr), ξ(1 + λ2)ν + e−2tλ2 µ

1 2 belongs to L (dγµ) ∩ L (dγµ). From the relation (4.18) and the fact that

2 e−tλ j (λs) Q (r, s) = H µ (r), ξ µ ξ(1 + λ2)ν + e−2tλ2

2 we deduce that for all g ∈ L (dγµ) and s ∈ [0, +∞[, we have

2 Z +∞ e−tλ j (λs) f ∗ (s) = g(r)H µ (r)dγ (r). ξ,g µ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e Applying Parseval’s equality, we get

2 Z +∞ e−tλ j (λs) f ∗ (s) = H (g)(λ) µ dγ (λ) ξ,g µ 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e 2 e−tλ = H H (g) (s), µ µ ξ(1 + λ2)ν + e−2tλ2 which implies that

2 e−tλ H (f ∗ )(λ) = H (g)(λ) , (4.22) µ ξ,g µ ξ(1 + λ2)ν + e−2tλ2

2 then for all g1, g2 ∈ L (dγµ),

2 2 ν −2tλ2 2 Z +∞ (1 + λ ) e Hµ(g1 − g2)(λ) ∗ ∗ f − f = dγµ(λ). ξ,g1 ξ,g2 2 ν 0 ξ(1 + λ2)ν + e−2tλ2

2 2 Applying again the fact that a + b > 2ab, a, b > 0, we obtain

+∞ 2 1 Z f ∗ − f ∗ |H (g − g )(λ)|2 dγ (λ) ξ,g1 ξ,g2 ν 6 µ 1 2 µ 4ξ 0 1 = kg − g k2 . 4ξ 1 2 2,µ 14 Weierstrass transform associated with the Hankel operator

µ Corollary 4.4 Let ν > µ + 1. For every f ∈ Hν ([0, +∞[) and g = Wµ,t(f), we have ∗ lim fξ,g − f = 0. ξ→0+ ν

∗ + Moreover, (fξ,g)ξ>0 converges uniformly to f as ξ → 0 .

µ Proof. Let f ∈ Hν ([0, +∞[) and g = Wµ,t(f). From Proposition 3.5, the 2 function g belongs to L (dγµ). Applying the relations (3.16) and (4.22), we obtain −ξ(1 + λ2)ν H (f)(λ) H (f ∗ − f)(λ) = µ . (4.23) µ ξ,g ξ(1 + λ2)ν + e−2tλ2 Consequently,

Z +∞ 2 2 2ν ∗ 2 ξ (1 + λ ) 2 ν 2 fξ,g − f = (1 + λ ) |Hµ(f)(λ)| dγµ(λ). ν 2 2 0 ξ(1 + λ2)ν + e−2tλ

Using the dominated convergence theorem and the fact that

2 2 3ν 2 ξ (1 + λ ) |Hµ(f)(λ)| 2 ν 2 2 6 (1 + λ ) |Hµ(f)(λ)| , ξ(1 + λ2)ν + e−2tλ2

µ and f ∈ Hν ([0, +∞[), we deduce that

∗ lim fξ,g − f = 0. ξ→0+ ν

1 2 From Remark 3.1, the function Hµ(f) belongs to L (dγµ) ∩ L (dγµ), then by inversion formula and the relation (4.23), we get

Z +∞ −ξ(1 + λ2)ν H (f)(λ) (f ∗ − f)(r) = µ j (λr)dγ (λ). ξ,g 2 ν −2tλ2 µ µ 0 ξ(1 + λ ) + e

So, for all r ∈ [0, +∞[,

Z +∞ ξ(1 + λ2)ν |H (f)(λ)| (f ∗ − f)(r) µ dγ (λ). ξ,g 6 2 ν −2tλ2 µ 0 ξ(1 + λ ) + e Again, by dominated convergence theorem and the fact that

ξ(1 + λ2)ν |H (f)(λ)| µ |H (f)(λ)| , ξ(1 + λ2)ν + e−2tλ2 6 µ we deduce that

∗ + sup (fξ,g − f)(r) −→ 0, as ξ −→ 0 . r∈[0,+∞[ S. Omri, & L. T. Rachdi 15

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Slim Omri Département de Mathématiques Appliquées, Institut Supérieurdes systèmes industriels de Gabès Rue Slaheddine El Ayoubi 6032 Gabès, Tunisia e-mail: [email protected] Lakhdar Tannech Rachdi Département de Mathématiques, Facultédes Sciences de Tunis El Manar II - 2092 Tunis, Tunisia e-mail: [email protected]