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
dr as
[0, +∞[ with respect to the measure µ
2 Γ(µ + 1)
Z +∞
1
Hµ (λ) = µ
f (r)jµ (λr)r2µ+1 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
Wµ,t (f )(r)
+∞
Et (r, s)f (s)
=
0
s2µ+1
ds,
+ 1)
2µ Γ(µ
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
`µ (u)(r, t) = ∂u
∂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
`µ =
2µ + 1 ∂
∂2
+
.
2
∂r
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
dr) be the Hilbert space of square integrable func2µ Γ(µ + 1)
r2µ+1
tions on [0, +∞[ with respect to the measure µ
dr, and h.|.iµ its inner
2 Γ(µ + 1)
product.
Let L2 ([0, +∞[,
For ν ∈ R, we consider the Sobolev type space Hµν ([0, +∞[), consisting of funcr2µ+1
tions f ∈ L2 ([0, +∞[, µ
dr) such that the function
2 Γ(µ + 1)
λ 7−→ (1 + λ2 )ν/2 Hµ (f )(λ),
r2µ+1
dr). Then for ν > µ+1, Hµν ([0, +∞[)
2µ Γ(µ + 1)
is the Hilbert space when equipped with the inner product
Z +∞
λ2µ+1
hf |giν =
dλ.
(1 + λ2 )ν Hµ (f )(λ)Hµ (g)(λ) µ
2 Γ(µ + 1)
0
belongs to the space L2 ([0, +∞[,
Moreover, the kernel
Z
Kν (r, s) =
0
+∞
jµ (λr)jµ (λs) λ2µ+1
dλ,
(1 + λ2 )ν 2µ Γ(µ + 1)
is a reproducing kernel of the space Hµν ([0, +∞[), where
jµ (z) =
+∞
X
(−1)n
z
2µ Γ(µ + 1)
J
(z)
=
Γ(µ
+
1)
( )2n ,
µ
µ
z
n!Γ(µ + n + 1) 2
n=0
z ∈ C,
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
Hµν ([0, +∞[) into L2 ([0, +∞[, µ
dr) and that for all f ∈ Hµν ([0, +∞[),
2 Γ(µ + 1)
kWµ,t (f )k2,µ 6 ||f ||ν .
Next, for ξ > 0, we define on the space Hµν ([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
We show that Hµν ([0, +∞[) equipped with the inner product h.|.iν,ξ , is a Hilbert
space and we exhibit a reproducing kernel, that is
Z +∞
λ2µ+1
jµ (λr)jµ (λs)
Kν,ξ (r, s) =
dλ.
2
ξ(1 + λ2 )ν + e−2tλ 2µ Γ(µ + 1)
0
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, +∞[, µ
dr), the
2 Γ(µ + 1)
infimum of
n
o
ξ||f ||2ν + ||g − Wµ,t (f )||22,µ ; f ∈ Hµν ([0, +∞[) ,
∗
is attained at one function fξ,g
, called the extremal function. We establish also,
the following estimates
• For all f ∈ Hµν ([0, +∞[) and g = Wµ,t (f ),
∗
− f ν = 0.
lim+ fξ,g
ξ→0
• For all f ∈ Hµν ([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
dγµ (r) =
r2µ+1
dr,
2µ Γ(µ + 1)
• Lp (dγµ ) , p ∈ [1, +∞], the space of measurable functions f on [0, +∞[
satisfying
Z
kf kp,µ =
+∞
p1
p
< +∞, if p ∈ [1, +∞[;
|f (r)| dγµ (r)
0
kf k∞,µ = ess sup |f (r)| < +∞,
if p = +∞.
r∈[0,+∞[
•
h.|.iµ the inner product on L2 (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
|r|→+∞
f (r) = 0.
4
Weierstrass transform associated with the Hankel operator
Let `µ be the Bessel operator defined on ]0, +∞[ by
`µ (u) = u00 +
2µ + 1 0
u,
r
then for all λ ∈ C, the following problem,

 `µ (u) = −λ2 u,
u(0) = 1,
 0
u (0) = 0,
admits a unique solution given by jµ (λ.), where
jµ (z) =
+∞
X
(−1)n
z
2µ Γ(µ + 1)
J
(z)
=
Γ(µ
+
1)
( )2n ,
µ
µ
z
n!Γ(µ + n + 1) 2
n=0
z ∈ C, (2.1)
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)
jµ (x) =
 πΓ(µ + (1/2))
cos x,
√
Z
1
(1 − t2 )µ−1/2 cos x t dt, if µ > −1/2;
0
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

p
Γ(µ + 1)


jµ ( r2 + s2 + 2rs cos θ)(sin θ)2µ dθ, if µ > −1/2;
Γ(µ + 1/2)Γ(1/2) 0
jµ (r)jµ (s) =

 j−1/2 (r + s) + j−1/2 (r − s) ,
if µ = −1/2.
2
Using the product formula, we will define and study the Hankel translation
operator and the convolution product.
Definition 2.1 1) For all r ∈ [0, +∞[, the Hankel translation operator τrµ is
defined on Lp (dγµ ) by
Z π p

Γ(µ + 1)


f ( r2 + s2 + 2rs cos θ)(sin θ)2µ dθ, if µ > −1/2;
µ
Γ(µ
+
1/2)Γ(1/2)
0
τr (f )(s) =

 f (r + s) + f (|r − s|) ,
if µ = −1/2.
2
S. Omri, & L. T. Rachdi
5
2) The convolution product of f, g ∈ L1 (dγµ ) is defined by
Z +∞
τrµ (f )(s)g(s)dγµ (s).
f ∗µ g(r) =
0
The following assumptions hold
• The product formula can be written
∀(r, s) ∈ [0, +∞[×[0, +∞[,
τrµ (jµ (λ.))(s) = jµ (λr)jµ (λs).
• For all f ∈ Lp (dγµ ), p ∈ [1, +∞], and for all r ∈ [0, +∞[ the function τrµ (f )
belongs to the space Lp (dγµ ) and
kτrµ (f )kp,µ 6 kf kp,µ .
(2.3)
• Let p, q, r ∈ [1, +∞] be such that 1/p + 1/q = 1 + 1/r. For all f ∈ Lp (dγµ ) and
g ∈ Lq (dγµ ), the function f ∗µ g belongs to Lr (dγµ ) and we have the following
Young inequality
kf ∗µ gkr,µ 6 kf kp,µ kgkq,µ .
(2.4)
• For all f ∈ L1 (dγµ ) and λ ∈ [0, +∞[, the function τλµ (f ) belongs to L1 (dγµ )
and we have
Z +∞
Z +∞
τλµ (f )(r)dγµ (r) =
f (r)dγµ (r).
(2.5)
0
0
Definition 2.2 The Hankel transform Hµ is defined on L1 (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
• For all f ∈ L1 (dγµ ) the function Hµ (f ) belongs to the space C∗,0 (R) and
kHµ (f )k∞,µ 6 kf k1,µ .
• For all f ∈ L1 (dγµ ) and r ∈ [0, ∞[
Hµ τrµ (f ) (λ) = jµ (rλ)Hµ (f )(λ).
(2.6)
• For f, g ∈ L1 (dγµ )
Hµ (f ∗µ g) = Hµ (f )Hµ (g).
Theorem 2.3 (Inversion formula for Hµ ) Let f ∈ L1 (dγµ ) such that Hµ (f ) ∈
L1 (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
to an isometric isomorphism from L2 (dγµ ) onto itself. In particular for all
f, g ∈ L2 (dγµ ), we have (Parseval equality)
Z +∞
Z +∞
f (r)g(r)dγµ (r) =
Hµ (f )(λ)Hµ (g)(λ)dγµ (λ).
0
0
1
Remark 2.5 i) Let f ∈ L (dγµ ) and g ∈ L2 (dγµ ), by the relation (2.4), the
function f ∗µ g belongs to L2 (dγµ ), moreover
Hµ (f ∗µ g) = Hµ (f )Hµ (g).
ii) For all f, g ∈ L2 (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 Hankel 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
• Hµν ([0, +∞[) the subspace of L2 (dγµ ) formed by the functions f , such that
the maps
λ 7−→ (1 + λ2 )ν/2 Hµ (f )(λ),
belongs to L2 (dγµ ).
• h.|.iν the inner product on Hµν ([0, +∞[) defined by
Z +∞
hf |giν =
(1 + λ2 )ν Hµ (f )(λ)Hµ (g)(λ)dγµ (λ).
0
• ||.||ν the norm of
Hµν ([0, +∞[)
defined by
p
||f ||ν = hf |f iν .
Remark 3.1 For ν > µ + 1, the function
λ 7−→
1
,
(1 + λ2 )ν/2
belongs to L2 (dγµ ). Hence for all f ∈ Hµν ([0, +∞[), the function Hµ (f ) belongs
to L1 (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)
dγµ (λ),
Kν (r, s) =
(1 + λ2 )ν
0
is a reproducing kernel of the space Hµν ([0, +∞[), that is
i) For all s ∈ [0, +∞[, the function
r 7−→ Kν (r, s),
belongs to Hµν ([0, +∞[).
ii) (The reproducing property) For all f ∈ Hµν ([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
λ 7−→
jµ (λs)
,
(1 + λ2 )ν
belongs to L1 (dγµ ) ∩ L2 (dγµ ). Then, the function Kν is well defined and by
Theorem 2.3, we have
Kν (r, s) = Hµ
j (λs) µ
(r).
(1 + λ2 )ν
By Plancherel theorem, it follows that for all s ∈ [0, +∞[, the function Kν (., s)
belongs to L2 (dγµ ), and we have
Hµ Kν (., s) (λ) =
jµ (λs)
.
(1 + λ2 )ν
Again, by the relation (2.2) and Remark 3.1, it follows that the function
λ 7−→ (1 + λ2 )ν/2 Hµ Kν (., s) (λ),
belongs to L2 (dγµ ).
ii) Let f be in Hµν ([0, +∞[). For every s ∈ [0, +∞[, we have
Z +∞
hf |Kν (., s)iν =
(1+λ2 )ν 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.
(3.8)
8
Weierstrass transform associated with the Hankel operator
The heat equation associated with the Hankel transform is given by
∂
u(r, t),
∂t
`µ u(r, t) =
(3.9)
where `µ is the Bessel operator defined above.
Let E be the kernel defined by
Z
E(r, t)
+∞
2
e−tλ jµ (rλ)dγµ (λ)
=
(3.10)
0
−r 2 /4t
=
e
.
(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
Et (r, s) = τrµ E(., t) (s)
(3.11)
2
=
2
e−(r +s )/4t
irs
jµ (
).
µ+1
(2t)
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, +∞[,
the function Et (r, .) belongs to L1 (dγµ ) and for all λ ∈ [0, +∞[, we have
2
Hµ Et (r, .) (λ) = e−tλ jµ (λr).
iii) From the relations (2.3), (2.5), (3.10) and (3.11), for all t > 0 and
s ∈ [0, +∞[, the function Et (., s) belongs to L1 (dγµ ) and we have
Z
+∞
Z
Et (r, s)dγµ (r) =
0
+∞
E(r, t)dγµ (r) = 1.
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 transform is defined on L2 (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].
Proposition 3.5 i) For all f ∈ L2 (dγµ ), the function Wµ,t (f ) solves the heat
equation (3.9), with the initial condition
lim Wµ,t (f ) = f,
t→0+
in L2 (dγµ ).
ii) For all t > 0 and ν > µ + 1, the transform Wµ,t is a bounded linear operator
from Hµν ([0, +∞[) into L2 (dγµ ) and for all f ∈ Hµν ([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 ∈
L2 (dγµ )
lim E(., t) ∗µ f = f
in L2 (dγµ ).
t→0
ii) From the relations (2.4) and (3.12), for all f ∈ L2 (dγµ ), we have
kWµ,t (f )k2,µ = kE(., t) ∗µ f k2,µ
6 kE(., t)k1,µ kf k2,µ
= kf k2,µ = kHµ (f )k2,µ 6 kf kν .
Notations. For all positive real numbers ξ, t and for ν > µ + 1, we denote
by
• h.|.iν,ξ , the inner product defined on the space Hµν ([0, +∞[) by
hf |giν,ξ = ξhf |giν + hWµ,t (f )|Wµ,t (g)iµ .
• Hµν,ξ ([0, +∞[), the space Hµν ([0, +∞[) equipped with the inner
product h.|.iν,ξ and the norm
||f ||2ν,ξ = ξ||f ||2ν + ||Wµ,t (f )||22,µ .
Then, we have the following main result [11, 15].
10
Weierstrass transform associated with the Hankel operator
Theorem 3.6 For all ξ, t > 0 and ν > µ + 1, the Hilbert space Hµν,ξ ([0, +∞[)
admits the following reproducing kernel,
Z +∞
jµ (rλ)jµ (sλ)
dγµ (λ),
Kν,ξ (r, s) =
ξ(1
+ λ2 )ν + e−2tλ2
0
that is
i) For all s ∈ [0, +∞[, the function r 7−→ Kν,ξ (r, s) belongs to Hµν,ξ ([0, +∞[).
ii) (The reproducing property.) For all f ∈ Hµν,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
.
2 6
2
ν
−2tλ
ξ(1
+
λ2 )ν
ξ(1 + λ ) + e
Then by the hypothesis ν > µ + 1, we deduce that for all s ∈ [0, +∞[, the
function
jµ (λs)
λ 7−→
,
ξ(1 + λ2 )ν + e−2tλ2
belongs to L1 (dγµ ) ∩ L2 (dγµ ), and by the Plancherel theorem, the function
jµ (λs)
(r)
(3.13)
r 7−→ Kν,ξ (r, s) = Hµ
ξ(1 + λ2 )ν + e−2tλ2
belongs to L2 (dγµ ), moreover the function
(1 + λ2 )ν/2 jµ (λs)
λ 7−→(1 + λ2 )ν/2 Hµ Kν,ξ (., s) (λ) =
,
ξ(1 + λ2 )ν + e−2tλ2
belongs to L2 (dγµ ).
This proves that for all s ∈ [0, +∞[, the function Kν,ξ (., s) belongs to the
space Hµν,ξ ([0, +∞[).
ii) Let f be in Hµν,ξ ([0, +∞[). By the relation (3.13), we get
!
Z +∞
jµ (λs)
2 ν
dγµ (λ).
hf |Kν,ξ (., s)iν =
(1 + λ ) Hµ (f )(λ) ×
ξ(1 + λ2 )ν + e−2tλ2
0
(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
Wµ,t Kν,ξ (., s) (r) = Hµ
jµ (λs)e−tλ
(r).
2
2
ν
−2tλ
ξ(1 + λ ) + e
(3.15)
S. Omri, & L. T. Rachdi
11
By the same way
2
= Hµ e−tλ Hµ (f ) (r).
Wµ,t (f )(r)
(3.16)
Thus,
2
hWµ,t (f )|Wµ,t
2
Kν,ξ (., s) iµ = hHµ e−tλ Hµ (f ) |Hµ
jµ (λs)e−tλ
iµ .
2
ξ(1 + λ2 )ν + e−2tλ
Using the Parseval formula, we get
2
2
hWµ,t (f )|Wµ,t Kν,ξ (., s) iµ = he−tλ Hµ (f )|
jµ (λs)e−tλ
iµ .
ξ(1 + λ2 )ν + e−2tλ2
(3.17)
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.
Theorem 4.1 Let ν > µ + 1, ξ > 0 and g ∈ L2 (dγµ ). Then there is a unique
∗
function fξ,g
∈ Hµν ([0, +∞[), where the infimum of
n
ξ||f ||2ν + ||g − Wµ,t (f )||22,µ ,
o
f ∈ Hµν ([0, +∞[) ,
∗
is given by
is attained. Moreover, the extremal function fξ,g
∗
fξ,g
(r)
Z
=
+∞
g(s)Qξ (r, s)dγµ (s),
(4.18)
0
where,
Z
Qξ (r, s) =
0
+∞
2
e−tλ jµ (λr)jµ (λs)
dγµ (λ).
ξ(1 + λ2 )ν + e−2tλ2
(4.19)
∗
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
∗
fξ,g
(s)
Z
= hg|Hµ
+∞
=
jµ (λs)e−tλ
iµ
2
ξ(1 + λ2 )ν + e−2tλ
+∞
Z
g(r)
0
Z
0
2
e−tλ jµ (λr)jµ (λs)
dγ
(λ)
dγµ (r)
µ
ξ(1 + λ2 )ν + e−2tλ2
+∞
=
g(r)Qξ (r, s)dγµ (r).
0
Corollary 4.2 Let ν > µ + 1, ξ > 0 and g ∈ L2 (dγµ ). The extremal function
∗
fξ,g
, satisfies the following inequality
Z +∞
∗ 2
2
fξ,g 6 Γ(ν − µ − 1)
er |g(r)|2 dγµ (r).
2µ+4
2,µ
ξ2
Γ(ν) 0
Proof. We have
∗
fξ,g
(s) =
Z
+∞
e−
r2
2
e
r2
2
g(r)Qξ (r, s)dγµ (r).
0
By Hölder’s inequality we get
Z +∞
Z
2
∗
|fξ,g
(s)|2 6
e−r dγµ (r)
0
+∞
0
2
2
er |g(r)|2 Qξ (r, s) dγµ (r) .
Integrating over [0, +∞[ with respect to the measure dγµ (s), we obtain
Z +∞
Z +∞ 2
∗ 2
2
2
−r 2
fξ,g 6
e
dγ
(r)
er |g(r)| kQξ (r, .)k2,µ dγµ (r) .
µ
2,µ
0
0
However, by the relation (4.19)
2
Qξ (r, s) = Hµ
e−tλ jµ (λr)
(s),
2
ξ(1 + λ2 )ν + e−2tλ
(4.20)
then by the Plancherel theorem
Z +∞
2
2
e−2tλ |jµ (λr)|
2
kQξ (r, .)k2,µ =
dγµ (λ).
ξ(1 + λ2 )ν + e−2tλ2 2
0
Since, a2 + b2 > 2ab, a, b > 0, and in virtue of the relation (2.2), it follows that
Z +∞
dγµ (λ)
1
2
kQξ (r, .)k2,µ 6
.
(4.21)
4ξ 0
(1 + λ2 )ν
We complete the proof by using the relations (4.20) and (4.21), and the fact
that
Z +∞
2
1
e−r dγµ (r) = µ+1 ,
2
0
and
Z +∞
dγµ (λ)
Γ(ν − µ − 1)
=
.
2 )ν
(1
+
λ
2µ+1 Γ(ν)
0
S. Omri, & L. T. Rachdi
13
Corollary 4.3 Let ν > µ + 1. For all g1 , g2 ∈ L2 (dγµ ), we have
∗
kg1 − g2 k2,µ
∗ fξ,g − fξ,g
√
6
.
1
2 ν
2 ξ
Proof. Let ν > µ + 1. For all r ∈ [0, +∞[, the function
2
e−tλ
λ 7−→
jµ (λr),
ξ(1 + λ2 )ν + e−2tλ2
belongs to L1 (dγµ ) ∩ L2 (dγµ ).
From the relation (4.18) and the fact that
2
Qξ (r, s) = Hµ
e−tλ jµ (λs)
(r),
2
ξ(1 + λ2 )ν + e−2tλ
we deduce that for all g ∈ L2 (dγµ ) and s ∈ [0, +∞[, we have
∗
fξ,g
(s)
Z
2
+∞
=
g(r)Hµ
0
e−tλ jµ (λs)
(r)dγµ (r).
2
ξ(1 + λ2 )ν + e−2tλ
Applying Parseval’s equality, we get
∗
fξ,g
(s)
2
+∞
e−tλ jµ (λs)
dγµ (λ)
ξ(1 + λ2 )ν + e−2tλ2
0
2
e−tλ
(s),
= Hµ Hµ (g)
2
ξ(1 + λ2 )ν + e−2tλ
Z
Hµ (g)(λ)
=
which implies that
2
∗
Hµ (fξ,g
)(λ)
= Hµ (g)(λ)
e−tλ
,
ξ(1 + λ2 )ν + e−2tλ2
then for all g1 , g2 ∈ L2 (dγµ ),
2 Z
∗
∗ fξ,g1 − fξ,g
=
2
ν
+∞
0
2
2
(1 + λ2 )ν e−2tλ Hµ (g1 − g2 )(λ)
dγµ (λ).
2
ξ(1 + λ2 )ν + e−2tλ2
Applying again the fact that a2 + b2 > 2ab, a, b > 0, we obtain
∗
∗ 2
fξ,g − fξ,g
1
2 ν
6
=
Z +∞
1
2
|Hµ (g1 − g2 )(λ)| dγµ (λ)
4ξ 0
1
2
kg1 − g2 k2,µ .
4ξ
(4.22)
14
Weierstrass transform associated with the Hankel operator
Corollary 4.4 Let ν > µ + 1. For every f ∈ Hµν ([0, +∞[) and g = Wµ,t (f ), we
have
∗
lim fξ,g
− f ν = 0.
ξ→0+
∗
Moreover, (fξ,g
)ξ>0 converges uniformly to f as ξ → 0+ .
Proof. Let f ∈ Hµν ([0, +∞[) and g = Wµ,t (f ). From Proposition 3.5, the
function g belongs to L2 (dγµ ). Applying the relations (3.16) and (4.22), we
obtain
−ξ(1 + λ2 )ν Hµ (f )(λ)
∗
.
(4.23)
Hµ (fξ,g
− f )(λ) =
ξ(1 + λ2 )ν + e−2tλ2
Consequently,
∗
fξ,g − f 2 =
ν
Z
+∞
ξ 2 (1 + λ2 )2ν
ξ(1 + λ2 )ν + e−2tλ2
0
2
2 ν
2 (1 + λ ) |Hµ (f )(λ)| dγµ (λ).
Using the dominated convergence theorem and the fact that
2
ξ 2 (1 + λ2 )3ν |Hµ (f )(λ)|
2
2 ν
6 (1 + λ ) |Hµ (f )(λ)| ,
2 2
2
ν
−2tλ
ξ(1 + λ ) + e
and f ∈ Hµν ([0, +∞[), we deduce that
∗
lim fξ,g
− f ν = 0.
ξ→0+
From Remark 3.1, the function Hµ (f ) belongs to L1 (dγµ ) ∩ L2 (dγµ ), then by
inversion formula and the relation (4.23), we get
∗
(fξ,g
− f )(r) =
Z
+∞
0
−ξ(1 + λ2 )ν Hµ (f )(λ)
jµ (λr)dγµ (λ).
ξ(1 + λ2 )ν + e−2tλ2
So, for all r ∈ [0, +∞[,
∗
(fξ,g − f )(r) 6
Z
0
+∞
ξ(1 + λ2 )ν |Hµ (f )(λ)|
dγµ (λ).
ξ(1 + λ2 )ν + e−2tλ2
Again, by dominated convergence theorem and the fact that
ξ(1 + λ2 )ν |Hµ (f )(λ)|
6 |Hµ (f )(λ)| ,
ξ(1 + λ2 )ν + e−2tλ2
we deduce that
sup
r∈[0,+∞[
∗
(fξ,g − f )(r) −→ 0, as ξ −→ 0+ .
S. Omri, & L. T. Rachdi
15
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Slim Omri
Département de Mathématiques Appliquées,
Institut Supérieur des systèmes industriels de Gabès
Rue Slaheddine El Ayoubi 6032 Gabès, Tunisia
e-mail: slim.omri@issig.rnu.tn
Lakhdar Tannech Rachdi
Département de Mathématiques,
Faculté des Sciences de Tunis
El Manar II - 2092 Tunis, Tunisia
e-mail: lakhdartannech.rachdi@fst.rnu.tn