International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 2 (2015), 145-152
http://www.etamaths.com

BEST APPROXIMATION OF THE DUNKL MULTIPLIER

OPERATORS Tk,`,m

FETHI SOLTANI

Abstract. We study some class of Dunkl multiplier operators Tk,`,m; and

we give for them an application of the theory of reproducing kernels to the

Tikhonov regularization, which gives the best approximation of the operators
Tk,`,m on a Hilbert spaces H

s
k`

.

1. Introduction

In this paper, we consider Rd with the Euclidean inner product 〈., .〉 and norm
|y| :=

√
〈y,y〉. For α ∈ Rd\{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd

orthogonal to α:

σαx := x−
2〈α,x〉
|α|2

α.

A finite set < ⊂ Rd\{0} is called a root system, if < ∩ R.α = {−α,α} and
σα< = < for all α ∈<. We assume that it is normalized by |α|2 = 2 for all α ∈<.
For a root system <, the reflections σα, α ∈ <, generate a finite group G. The
Coxeter group G is a subgroup of the orthogonal group O(d). All reflections in
G, correspond to suitable pairs of roots. For a given β ∈ Rd\

⋃
α∈<Hα, we fix

the positive subsystem <+ := {α ∈ < : 〈α,β〉 > 0}. Then for each α ∈ < either
α ∈<+ or −α ∈<+.

Let k,` : < → C be two multiplicity functions on < (a functions which are
constants on the orbits under the action of G). As an abbreviation, we introduce
the index γk :=

∑
α∈<+ k(α) and γ` :=

∑
α∈<+ `(α).

Throughout this paper, we will assume that k(α),`(α) ≥ 0 for all α ∈ <, and
γ` ≥ γk. Moreover, let wk denote the weight function wk(x) :=

∏
α∈<+ |〈α,x〉|

2k(α),

for all x ∈ Rd, which is G-invariant and homogeneous of degree 2γk.
Let ck be the Mehta-type constant given by

ck :=

(∫
Rd
e−|x|

2/2wk(x)dx

)−1
.

2010 Mathematics Subject Classification. 42B10; 42B15; 46E35.
Key words and phrases. Hilbert spaces; Dunkl multiplier operators; Tikhonov regularization;

extremal functions.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

145



146 FETHI SOLTANI

We denote by µk the measure on Rd given by dµk(x) := ckwk(x)dx; and by Lp(µk),
1 ≤ p ≤∞, the space of measurable functions f on Rd, such that

‖f‖Lp(µk) :=
(∫

Rd
|f(x)|pdµk(x)

)1/p
< ∞, 1 ≤ p < ∞,

‖f‖L∞(µk) := ess sup
x∈Rd

|f(x)| < ∞.

For f ∈ L1(µk) the Dunkl transform is defined (see [2]) by

Fk(f)(y) :=
∫
Rd
Ek(−ix,y)f(x)dµk(x), y ∈ Rd,

where Ek(−ix,y) denotes the Dunkl kernel (for more details, see the next section).
Let s > 0. We consider the Hilbert Hsk` consisting of functions f ∈ L

2(µ`) such

that es|z|
2/2F`(f) ∈ L2(µk). The space Hsk` is endowed with the inner product

〈f,g〉Hs
k`

:=

∫
Rd
es|z|

2

F`(f)(z)F`(g)(z)dµk(z).

Let m be a function in L2(µk). The Dunkl multiplier operators Tk,`,m, are
defined for f ∈ Hsk` by

Tk,`,mf(x,a) := F−1k (m(a.)F`(f))(x), (x,a) ∈ K := R
d × (0,∞).

These operators are studied in [14] where the author established some applications
(Calderón’s reproducing formulas, best approximation formulas, extremal function-
s....). In particular, when k = ` these operators are studied in [13].

For m ∈ L2(µk) satisfying the admissibility condition:
∫∞

0
|m(ax)|2 da

a
= 1,

a.e. x ∈ Rd, then the operators Tk,`,m satisfy, for f ∈ Hsk`:
‖Tk,`,mf‖2L2(Ωk) = ‖F`(f)‖

2
L2(µk)

,

where Ωk is the measure on K given by dΩk(x,a) := daa dµk(x).
Building on the ideas of Matsuura et al. [5], Saitoh [9, 11] and Yamada et al.

[18], and using the theory of reproducing kernels [8], we give best approximation
of the operator Tk,`,m on the Hilbert spaces H

s
k`. More precisely, for all λ > 0,

g ∈ L2(Ωk), the infimum

inf
f∈Hs

k`

{
λ‖f‖2Hs

k`
+ ‖g −Tk,`,mf‖2L2(Ωk)

}
,

is attained at one function f∗λ,g, called the extremal function, and given by

F∗λ,g(y) =

∫
Rd

E`(iy,z)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
dµ`(z).

Next we show for F∗λ,g the following properties.

(i) ‖F∗λ,g‖Hsk` ≤
1

2
√
λ
‖g‖L2(Ωk).

(ii) Tk,`,mF
∗
λ,g(y,a) =

∫
Rd

m(az)Ek(iy,z)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
dµk(z).

In the Dunkl setting, the extremal functions are studied in several directions
[12, 13, 14, 15, 16].

This paper is organized as follows. In section 2 we define and study the Dunkl
multiplier operators Tk,`,m on the Hilbert space H

s
k`. The last section of this paper is



BEST APPROXIMATION OF THE DUNKL MULTIPLIER OPERATORS Tk,`,m 147

devoted to give an application of the theory of reproducing kernels to the Tikhonov
regularization, which gives the best approximation of the operators Tk,`,m on the
Hilbert space Hsk`.

2. Dunkl type multiplier operators

The Dunkl operators Dj; j = 1, ...,d, on Rd associated with the finite reflection
group G and multiplicity function k are given, for a function f of class C1 on Rd,
by

Djf(x) :=
∂

∂xj
f(x) +

∑
α∈<+

k(α)αj
f(x) −f(σαx)
〈α,x〉

.

For y ∈ Rd, the initial problem Dju(.,y)(x) = yju(x,y), j = 1, ...,d, with
u(0,y) = 1 admits a unique analytic solution on Rd, which will be denoted by
Ek(x,y) and called Dunkl kernel [1, 3]. This kernel has a unique analytic extension
to Cd ×Cd (see [7]). In our case (see [1, 2]),

|Ek(ix,y)| ≤ 1, x,y ∈ Rd. (2.1)
The Dunkl kernel gives rise to an integral transform, which is called Dunkl

transform on Rd, and was introduced by Dunkl in [2], where already many basic
properties were established. Dunkl’s results were completed and extended later by
De Jeu [3]. The Dunkl transform of a function f in L1(µk), is defined by

Fk(f)(y) :=
∫
Rd
Ek(−ix,y)f(x)dµk(x), y ∈ Rd.

We notice that F0 agrees with the Fourier transform F that is given by

F(f)(y) := (2π)−d/2
∫
Rd
e−i〈x,y〉f(x)dx, x ∈ Rd.

Some of the properties of Dunkl transform Fk are collected bellow (see [2, 3]).
Theorem 2.1 (i) L1 −L∞-boundedness. For all f ∈ L1(µk), Fk(f) ∈ L∞(µk) and

‖Fk(f)‖L∞(µk) ≤‖f‖L1(µk).

(ii) Inversion theorem. Let f ∈ L1(µk), such that Fk(f) ∈ L1(µk). Then

f(x) = Fk(Fk(f))(−x), a.e. x ∈ Rd.
(iii) Plancherel theorem. The Dunkl transform Fk extends uniquely to an iso-

metric isomorphism of L2(µk) onto itself. In particular,

‖Fk(f)‖L2(µk) = ‖f‖L2(µk).

Let s > 0. We define the Hilbert space Hsk`, as the set of all f ∈ L
2(µ`) such

that es|z|
2/2F`(f) ∈ L2(µk). The space Hsk` provided with the inner product

〈f,g〉Hs
k`

:=

∫
Rd
es|z|

2

F`(f)(z)F`(g)(z)dµk(z),

and the norm ‖f‖Hs
k`

=
√
〈f,f〉Hs

k`
. The space Hsk` satisfies the following proper-

ties.
(i) The Hsk` has the reproducing kernel

hsk`(x,y) =
c`
ck

∫
Rd
e−s|z|

2

E`(ix,z)E`(−iy,z)w`−k(z)dµ`(z).



148 FETHI SOLTANI

If k = `, then hskk is the Dunkl-type heat kernel [6, 12] and this kernel is given by

hskk(x,y) =
1

(2s)γk+d/2
e−(|x|

2+|y|2)/4sEk

( x
√

2s
,
y
√

2s

)
.

(ii) The space Hsk` is continuously contained in L
2(µ`) and

‖f‖2L2(µ`) ≤
c`
ck

(2
e

)γ`−γk(γ` −γk
s

)γ`−γk
‖f‖2Hs

k`
.

(iii) If f ∈ Hsk` then F`(f) ∈ L
1(µ`) and ‖F`(f)‖L1(µ`) ≤ Ck,`‖f‖Hsk` , where

Ck,` =

(
c`
ck

∫
Rd
e−s|z|

2

w`−k(z)dµ`(z)

)1/2
. (2.2)

(iv) If f ∈ Hsk`, then F`(f) ∈ L
1 ∩L2(µ`) and

f(x) =

∫
Rd
E`(ix,z)F`(f)(z)dµ`(z), a.e. x ∈ Rd.

Let λ > 0. We denote by 〈., .〉λ,Hs
k`

the inner product defined on the space Hsk`
by

〈f,g〉λ,Hs
k`

:= λ〈f,g〉Hs
k`

+ 〈F`(f),F`(g)〉L2(µk), (2.3)

and the norm ‖f‖λ,Hs
k`

:=
√
〈f,f〉λ,Hs

k`
. On Hsk` the two norms ‖.‖Hsk` and ‖.‖λ,Hsk`

are equivalent. This (Hsk`,〈., .〉λ,Hsk` ) is a Hilbert space with reproducing kernel
given by

Ksk`(x,y) =
c`
ck

∫
Rd

E`(ix,z)E`(−iy,z)
1 + λes|z|

2 w`−k(z)dµ`(z). (2.4)

Let m be a function in L2(µk). The Dunkl multiplier operators Tk,`,m, are
defined for f ∈ Hsk` by

Tk,`,mf(x,a) := F−1k (m(a.)F`(f))(x), (x,a) ∈ K. (2.5)

We denote by Ωk the measure on K given by dΩk(x,a) := daa dµk(x); and by
L2(Ωk), the space of measurable functions F on K, such that

‖F‖L2(Ωk) :=
(∫

Rd

∫ ∞
0

|F(x,a)|2dΩk(x,a)
)1/2

< ∞.

Let m be a function in L2(µk) satisfying the admissibility condition∫ ∞
0

|m(ax)|2
da

a
= 1, a.e. x ∈ Rd. (2.6)

Then from Theorem 2.1 (iii) , for f ∈ Hsk`, we have

‖Tk,`,mf‖L2(Ωk) = ‖F`(f)‖L2(µk) ≤‖f‖Hsk`. (2.7)



BEST APPROXIMATION OF THE DUNKL MULTIPLIER OPERATORS Tk,`,m 149

3. Extremal functions for the operators Tk,`,m

In this section, by using the theory of extremal function and reproducing kernel
of Hilbert space [8, 9, 10, 11] we study the extremal function associated to the
Dunkl multiplier operators Tk,`,m. In the particular case when k = ` this function
is studied in [16, 17]. The main result of this section can be stated as follows.
Theorem 3.1. Let m ∈ L2(µk) satisfying (2.6). For any g ∈ L2(Ωk) and for any
λ > 0, there exists a unique function F∗λ,g, where the infimum

inf
f∈Hs

k`

{
λ‖f‖2Hs

k`
+ ‖g −Tk,`,mf‖2L2(Ωk)

}
(3.1)

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

F∗λ,g(y) =

∫
Rd

∫ ∞
0

g(x,a)Qs(x,y,a)dΩk(x,a),

where

Qs(x,y,a) =

∫
Rd

m(az)Ek(−ix,z)E`(iy,z)
1 + λes|z|

2 dµ`(z).

Proof. Let s,λ > 0. Since m ∈ L2(µk) and satisfying (2.6), then by (2.7), the
inner product 〈., .〉λ,Hs

k`
defined by (2.3) is written by

〈f,g〉λ,Hs
k`

= λ〈f,g〉Hs
k`

+ 〈Tk,`,mf,Tk,`,mg〉L2(Ωk).

Then, the existence and unicity of the extremal function F∗λ,g satisfying (3.1) is

obtained in [4, 5, 10]. Especially, F∗η,g is given by the reproducing kernel of H
s
k`

with ‖.‖λ,Hs
k`

norm as

F∗λ,g(y) = 〈g,Tk,`,m(K
s
k`(.,y))〉L2(Ωk), (3.2)

where Ksk` is the kernel given by (2.4). Then, we obtain the result by Theorem 2.1
(ii) and the fact that

F`(Ksk`(.,y))(z) =
c`
ck

E`(−iy,z)
1 + λes|z|

2 w`−k(z), z ∈ R
d. (3.3)

�

Theorem 3.2. Let λ > 0 and g ∈ L2(Ωk). The extremal function F∗λ,g satisfies

(i) |F∗λ,g(y)| ≤
Ck,`

2
√
λ
‖g‖L2(Ωk),

where Ck,` is the constant given by (2.2).

(ii) ‖F∗λ,g‖
2
L2(µ`)

≤
Dk,`
λ
‖m‖2L2(µk)

∫
Rd

∫ ∞
0

|g(x,a)|2
e(|x|

2+a2)/2

a2γk+d+1
dΩk(x,a),

where

Dk,` =
ck
√
π

4c`
√

2a2γk+d

(2
e

)γ`−γk(γ` −γk
s

)γ`−γk
.

Proof. (i) From (2.7) and (3.2), we have

|F∗λ,g(y)| ≤ ‖g‖L2(Ωk)‖Tk,`,m(K
s
k`(.,y))‖L2(Ωk)

≤ ‖g‖L2(Ωk)‖F`(K
s
k`(.,y))‖L2(µk).

Then, by (3.3) we deduce

|F∗λ,g(y)| ≤ ‖g‖L2(Ωk)
(
c`
ck

∫
Rd

w`−k(z)dµ`(z)

[1 + λes|z|
2
]2

)1/2
.



150 FETHI SOLTANI

Using the fact that [1 + λes|z|
2

]2 ≥ 4λes|z|
2

, we obtain the result.
(ii) We write

F∗λ,g(y) =

∫
Rd

∫ ∞
0

√
ae−(|x|

2+a2)/4 e
(|x|2+a2)/4
√
a

g(x,a)Qs(x,y,a)dΩk(x,a).

Applying Hölder’s inequality, we obtain

|F∗λ,g(y)|
2 ≤

√
π

2

∫
Rd

∫ ∞
0

|g(x,a)|2
e(|x|

2+a2)/2

a
|Qs(x,y,a)|2dΩk(x,a).

Thus and from Fubini-Tonnelli’s theorem, we get

‖F∗λ,g‖
2
L2(µ`)

≤
√
π

2

∫
Rd

∫ ∞
0

|g(x,a)|2
e(|x|

2+a2)/2

a
‖Qs(x,.,a)‖2L2(µ`)dΩk(x,a).

(3.4)

The function z → m(az)Ek(−ix,z)
1+λes|z|

2 belongs to L
1 ∩L2(µ`), then by Theorem 2.1 (ii),

Qs(x,y,a) = F−1`
(m(az)Ek(−ix,z)

1 + λes|z|
2

)
(y).

Thus, by Theorem 2.1 (iii) we deduce that

‖Qs(x,.,a)‖2L2(µ`) =
∫
Rd
|F`(Qs(x,.,a))(z)|2dµ`(z) ≤

∫
Rd

|m(az)|2dµ`(z)
[1 + λes|z|

2
]2

.

Then

‖Q(x,.,a)‖2L2(µ`) ≤
ck

4λc`

∫
Rd
e−s|z|

2

|m(az)|2w`−k(z)dµk(z)

≤
ck

4λc`a2γk+d

(2
e

)γ`−γk(γ` −γk
s

)γ`−γk
‖m‖2L2(µk).

From this inequality we deduce the result. �

Theorem 3.3. Let s,λ > 0. For every g ∈ L2(Ωk), we have

(i) F∗λ,g(y) =

∫
Rd

E`(iy,z)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
dµ`(z).

(ii) F`(F∗λ,g)(z) =
1

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
.

(iii) ‖F∗λ,g‖Hsk` ≤
1

2
√
λ
‖g‖L2(Ωk).

Proof. (i) From (3.2) we have

F∗λ,g(y) =

∫
Rd

∫ ∞
0

g(x,b)Tk,`,m(K
s
k`(.,y))(x,b)dΩk(x,b).

Since∫
Rd

∫ ∞
0

|g(x,b)Tk,`,m(Ksk`(.,y))(x,b)|dΩk(x,b) ≤‖g‖L2(Ωk)‖F`(K
s
k`(.,y))‖L2(µk) < ∞,



BEST APPROXIMATION OF THE DUNKL MULTIPLIER OPERATORS Tk,`,m 151

then, by Fubini’s theorem, Theorem 2.1 (iii) and (3.3) we obtain

F∗λ,g(y) =

∫ ∞
0

∫
Rd
g(x,b)Tk,`,m(K

s
k`(.,y))(x,b)dµk(x)

db

b

=

∫ ∞
0

∫
Rd
m(bz)Fk(g(.,b))(z)F`(Ksk`(.,y))(z)dµk(z)

db

b

=

∫ ∞
0

∫
Rd

m(bz)Fk(g(.,b))(z)E`(iy,z)
1 + λes|z|

2 dµ`(z)
db

b
.

Since ∫ ∞
0

∫
Rd

∣∣∣m(bz)Fk(g(.,b))(z)E`(iy,z)
1 + λes|z|

2

∣∣∣dµ`(z) db
b
≤
Ck,`

2
√
λ
‖g‖L2(Ωk) < ∞,

then, by Fubini’s theorem we deduce that

F∗λ,g(y) =

∫
Rd

E`(iy,z)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
dµ`(z).

(ii) The function z → 1
1+λes|z|

2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
belongs to L1 ∩

L2(µ`). Then by Theorem 2.1 (ii) and (iii), it follows that F
∗
λ,g belongs to L

2(µ`),
and

F`(F∗λ,g)(z) =
1

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
.

(iii) From (ii), Hölder’s inequality and (2.6) we have

|F`(F∗η,g)(z)|
2 ≤

1

[1 + ηes|z|
2
]2

[∫ ∞
0

|Fk(g(.,b))(z)|2
db

b

]
.

Thus,

‖F∗λ,g‖
2
Hs

k`
≤

∫
Rd

es|z|
2

[1 + λes|z|
2
]2

[∫ ∞
0

|Fk(g(.,b))(z)|2
db

b

]
dµk(z)

≤
1

4λ

∫
Rd

[∫ ∞
0

|Fk(g(.,b))(z)|2
db

b

]
dµk(z) =

1

4λ
‖g‖2L2(Ωk),

which ends the proof. �

Theorem 3.4. Let s,λ > 0. For every g ∈ L2(Ωk), we have

Tk,`,mF
∗
λ,g(y,a) =

∫
Rd

m(az)Ek(iy,z)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
dµk(z).

Proof. From (2.5) and Theorem 3.3 (ii), we have

Tk,`,mF
∗
λ,g(y,a) = F

−1
k

(
m(az)

1 + λes|z|
2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

])
(y).

The function z → m(az)
1+λes|z|

2

[∫ ∞
0

m(bz)Fk(g(.,b))(z)
db

b

]
belongs to L1(µk). Then

by Theorem 2.1 (ii), we obtain the result. �

Acknowledgments

The Author is partially supported by the DGRST research project LR11ES11
and CMCU program 10G/1503



152 FETHI SOLTANI

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