Best approximation of the Dunkl Multiplier Operators Tk,ℓ,m

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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 spacesHk`s.

1. Introduction

In this paper, we considerRd with the Euclidean inner product h., .iand norm

|y|:=phy, yi. Forα∈Rd\{0}, letσαbe the reflection in the hyperplaneHα⊂Rd

orthogonal toα:

σαx:=x−

2hα, xi |α|2 α.

A finite set < ⊂ Rd\{0} is called a root system, if < ∩R.α = {−α, α} and σα<=<for allα∈ <. We assume that it is normalized by|α|2= 2 for allα∈ <.

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_\S

α∈<Hα, we fix

the positive subsystem <+ :={α∈ < : hα, βi>0}. Then for eachα∈ < 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 indexγk:=Pα∈<+k(α) andγ`:=

P

α∈<+`(α).

Throughout this paper, we will assume that k(α), `(α)≥0 for all α∈ <, and

γ`≥γk. Moreover, letwk denote the weight functionwk(x) :=Qα∈<+|hα, xi|

2k(α)_,

for allx∈Rd, which isG-invariant and homogeneous of degree 2γk.

Letck be the Mehta-type constant given by

ck:=

Z

Rd

e−|x|2/2wk(x)dx

−1

.

2010Mathematics 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.

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We denote byµkthe measure onRdgiven by dµk(x) :=ckwk(x)dx; and byLp(µk),

1≤p≤ ∞, the space of measurable functionsf onRd, such that

kfkLp₍_µ k):=

Z

Rd

|f(x)|pdµk(x)

1/p

<∞, 1≤p <∞,

kfkL∞₍_µ

k):= ess sup

x∈Rd

|f(x)|<∞.

Forf ∈L1₍_µ

k) the Dunkl transform is defined (see [2]) by

Fk(f)(y) :=

Z

Rd

Ek(−ix, y)f(x)dµk(x), y∈Rd,

whereEk(−ix, y) denotes the Dunkl kernel (for more details, see the next section).

Lets >0. We consider the Hilbert Hs

k`consisting of functionsf ∈L2(µ`) such

thates|z|2_/₂

F`(f)∈L2(µk). The spaceHk`s is endowed with the inner product

hf, giHs k` :=

Z

Rd

es|z|2F`(f)(z)F`(g)(z)dµk(z).

Let m be a function in L2₍_µ

k). The Dunkl multiplier operators Tk,`,m, are

defined forf ∈Hs k`by

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

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

For m ∈ L2₍_µ

k) satisfying the admissibility condition:

R∞

0 |m(ax)| 2da

a = 1,

a.e. x∈Rd, then the operatorsTk,`,m satisfy, for f ∈Hk`s:

kTk,`,mfk2L2_(Ω

k)=kF`(f)k

2

L2₍_µ

k),

where Ωk is the measure onKgiven by dΩk(x, a) := d_aadµk(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 Hk`s. More precisely, for all λ > 0,

g∈L2_(Ω

k), the infimum

inf

f∈Hs k`

n λkfk2

Hs

k`+kg−Tk,`,mfk

2

L2_(Ω

k) o

,

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

F_λ,g∗ (y) =

Z

Rd

E`(iy, z)

1 +λes|z|2

Z ∞

0

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

db b

dµ`(z).

Next we show forF_λ,g∗ the following properties.

(i)kF_λ,g∗ kHs k` ≤

1

2√λkgkL2(Ωk).

(ii)Tk,`,mFλ,g∗ (y, a) =

Z

Rd

m(az)Ek(iy, z)

1 +λes|z|2

Z ∞

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].

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k,`,m

devoted to give an application of the theory of reproducing kernels to the Tikhonov regularization, which gives the best approximation of the operatorsTk,`,m on the

Hilbert spaceH_k`s.

2. _{Dunkl type multiplier operators}

The Dunkl operatorsDj;j = 1, ..., d, onRd associated with the finite reflection

group Gand multiplicity functionkare given, for a function f of classC1 onRd,

by

Djf(x) :=

∂ ∂xj

f(x) + X

α∈<+

k(α)αj

f(x)−f(σαx) hα, xi .

For y ∈ _Rd_{, the initial problem} _D

ju(., 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

toCd×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 functionf inL1₍_µ

k), is defined by

Fk(f)(y) :=

Z

Rd

Ek(−ix, y)f(x)dµk(x), y∈Rd.

We notice thatF0 agrees with the Fourier transformF that is given by

F(f)(y) := (2π)−d/2

Z

Rd

e−ihx,yif(x)dx, x∈Rd.

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

kFk(f)kL∞₍_µ

k)≤ kfkL1(µk).

(ii) Inversion theorem. Let f ∈L1(µk), such thatFk(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₍_µ

k)onto itself. In particular,

kFk(f)kL2₍_µ

k)=kfkL2(µk).

Let s >0. We define the Hilbert space H_k`s, as the set of allf ∈L2(µ`) such

thates|z|2_/₂

F`(f)∈L2(µk). The spaceHk`s provided with the inner product

hf, giHs k` :=

Z

Rd

es|z|2F`(f)(z)F`(g)(z)dµk(z),

and the norm kfkHs k` =

q

hf, fiHs

k`. The spaceH

s

k`satisfies the following

proper-ties.

(i) TheHs

k`has the reproducing kernel

hs_k`(x, y) = c`

ck

Z

Rd

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Ifk=`, then hs_kk is the Dunkl-type heat kernel [6, 12] and this kernel is given by

hs_kk(x, y) = 1 (2s)γk+d/2e

−(|x|2₊_|_y_|2₎_/₄_s

Ek

x

√

2s, y

√

2s

.

(ii) The spaceH_k`s is continuously contained inL2(µ`) and

kfk2

L2₍_µ_`₎≤

c`

ck

2 e

γ`−γkγ_`−γ_k s

γ`−γk

kfk2

Hs k`.

(iii) Iff ∈Hs

k`thenF`(f)∈L1(µ`) andkF`(f)kL1₍_µ_`₎≤Ck,`kfkHs

k`, where

Ck,`=

_c

`

ck

Z

Rd

e−s|z|2w`−k(z)dµ`(z)

1/2

. (2.2)

(iv) Iff ∈H_k`s, then F`(f)∈L1∩L2(µ`) and

f(x) =

Z

Rd

E`(ix, z)F`(f)(z)dµ`(z), a.e. x∈Rd.

Letλ >0. We denote byh., .iλ,Hs

k` the inner product defined on the spaceH

s k`

by

hf, giλ,Hs

k` :=λhf, giHk`s +hF`(f),F`(g)iL2(µk), (2.3)

and the normkfkλ,Hs k` :=

q

hf, fiλ,Hs

k`. OnH

s

k`the two normsk.kHs

k` andk.kλ,Hk`s

are equivalent. This (Hs

k`,h., .iλ,Hs

k`) is a Hilbert space with reproducing kernel

given by

K_k`s(x, y) = c`

ck

Z

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₍_µ

k). The Dunkl multiplier operators Tk,`,m, are

defined forf ∈Hs k`by

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

We denote by Ωk the measure on K given by dΩk(x, a) := d_aadµk(x); and by

L2(Ωk), the space of measurable functionsF onK, such that

kFkL2_(Ω_k₎:=

Z

Rd

Z ∞

0

|F(x, a)|2_dΩ

k(x, a)

1/2 <∞.

Letmbe a function inL2₍_µ

k) satisfying the admissibility condition

Z ∞

0

|m(ax)|2da

a = 1, a.e. x∈R

d_. ₍₂_.₆₎

Then from Theorem 2.1 (iii) , forf ∈Hs

k`, we have

kTk,`,mfkL2_(Ω

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k,`,m

3. Extremal functions for the operatorsTk,`,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 operatorsTk,`,m. In the particular case whenk=`this function

is studied in [16, 17]. The main result of this section can be stated as follows.

Theorem 3.1. Let m∈L2₍_µ

k)satisfying (2.6). For any g∈L2(Ωk)and for any

λ >0, there exists a unique functionF_λ,g∗ , where the infimum

inf

f∈Hs k`

n λkfk2

Hs

k`+kg−Tk,`,mfk

2

L2(Ωk) o

(3.1)

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

F_λ,g∗ (y) =

Z

Rd

Z ∞

0

g(x, a)Qs(x, y, a)dΩk(x, a),

where

Qs(x, y, a) =

Z

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 producth., .iλ,Hs

k` defined by (2.3) is written by

hf, giλ,Hs

k` =λhf, giHk`s +hTk,`,mf, Tk,`,mgiL2(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 Hs

k`

withk.kλ,Hs

k` norm as

F_λ,g∗ (y) =hg, Tk,`,m(Kk`s (., y))iL2_(Ω

k), (3.2)

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

F`(Kk`s(., y))(z) =

c`

ck

E`(−iy, z)

1 +λes|z|2w`−k(z), z∈R

d_. ₍₃_.₃₎

Theorem 3.2. Let λ >0 andg∈L2(Ωk). The extremal functionFλ,g∗ satisfies

(i)|F_λ,g∗ (y)| ≤ Ck,`

2√λkgkL2(Ωk),

whereCk,` is the constant given by (2.2).

(ii)kF_λ,g∗ k2

L2₍_µ

`)≤ Dk,`

λ kmk

2

L2₍_µ

k) Z

Rd

Z ∞

0

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

2₊_a2₎_/₂

a2γk+d+1 dΩk(x, a),

where

Dk,`=

ck √

π

4c` √

2a2γk+d 2

e

γ`−γk .

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

|F_λ,g∗ (y)| ≤ kgkL2_(Ω_k₎kTk,`,m(Kk`s(., y))kL2_(Ω_k₎ ≤ kgkL2_(Ω_k₎kF`(Kk`s(., y))kL2₍_µ_k₎.

Then, by (3.3) we deduce

|F_λ,g∗ (y)| ≤ kgkL2_(Ω

k) _c

`

ck

Z

Rd

w`−k(z)dµ`(z)

[1 +λes|z|2

]2

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Using the fact that [1 +λes|z|2]2≥4λes|z|2, we obtain the result. (ii) We write

F_λ,g∗ (y) =

Z

Rd

Z ∞

0

√

ae−(|x|2+a2)/4e

(|x|2₊_a2₎_/₄ √

a g(x, a)Qs(x, y, a)dΩk(x, a).

Applying H¨older’s inequality, we obtain

|F_λ,g∗ (y)|2_≤

r π

2

Z

Rd

Z ∞

0

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

2₊_a2₎_/₂

a |Qs(x, y, a)|

2_dΩ

k(x, a).

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

kF_λ,g∗ k2

L2₍_µ

`)≤ r

π

2

Z

Rd

Z ∞

0

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

2₊_a2₎_/₂

a kQs(x, ., a)k

2

L2₍_µ

`)dΩk(x, a).

(3.4)

The functionz→ m(az)Ek(−ix,z)

1+λes|z|2 belongs toL

1_∩_L2₍_µ

`), then by Theorem 2.1 (ii),

Qs(x, y, a) =F`−1

m(az)E_k(−ix, z)

1 +λes|z|2

(y).

Thus, by Theorem 2.1 (iii) we deduce that

kQs(x, ., a)k2L2₍_µ

`)= Z

Rd

|F`(Qs(x, ., a))(z)|2dµ`(z)≤

Z

Rd

|m(az)|2_d_µ

`(z)

[1 +λes|z|2

]2 .

Then

kQ(x, ., a)k2

L2(µ`) ≤ ck

4λc`

Z

Rd

e−s|z|2|m(az)|2_w

`−k(z)dµk(z)

≤ ck

4λc`a2γk+d 2

e

γ`−γk

kmk2

L2₍_µ

k).

From this inequality we deduce the result.

Theorem 3.3. Let s, λ >0. For everyg∈L2_(Ω

k), we have

(i)Fλ,g∗ (y) =

Z

Rd

E`(iy, z)

1 +λes|z|2

Z ∞

0

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

db b

dµ`(z).

(ii)F`(Fλ,g∗ )(z) =

1 1 +λes|z|2

Z ∞

0

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

db b

.

(iii)kF_λ,g∗ kHs k` ≤

1

2√λkgkL

2_(Ω

k).

Proof. (i) From (3.2) we have

F_λ,g∗ (y) =

Z

Rd

Z ∞

0

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

Since

Z

Rd

Z ∞

0

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k,`,m

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

F_λ,g∗ (y) =

Z ∞

0

Z

Rd

g(x, b)Tk,`,m(Kk`s(., y))(x, b)dµk(x)

db b = Z ∞ 0 Z Rd

m(bz)Fk(g(., b))(z)F`(Kk`s(., y))(z)dµk(z)

db b = Z ∞ 0 Z Rd

m(bz)Fk(g(., b))(z)E`(iy, z)

1 +λes|z|2 dµ`(z)

db b . Since Z ∞ 0 Z Rd

m(bz)Fk(g(., b))(z)E`(iy, z)

1 +λes|z|2

dµ`(z)

db b ≤

Ck,`

2√λkgkL2(Ωk)<∞,

then, by Fubini’s theorem we deduce that

F_λ,g∗ (y) =

Z

Rd

E`(iy, z)

1 +λes|z|2

Z ∞

0

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

db b

dµ`(z).

(ii) The function z → 1 1+λes|z|2

Z ∞

0

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

db b

belongs to L1 _∩

L2₍_µ

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

and

F`(Fλ,g∗ )(z) =

1 1 +λes|z|2

Z ∞

0

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

db b

.

(iii) From (ii), H¨older’s inequality and (2.6) we have

|F`(Fη,g∗ )(z)|

2_≤ 1

[1 +ηes|z|2

]2

Z ∞

0

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

db b

.

Thus,

kFλ,g∗ k

2

Hs k` ≤

Z

Rd

es|z|2

[1 +λes|z|2

]2

Z ∞

0

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

db b

dµk(z)

≤ 1 4λ Z Rd Z ∞ 0

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

db b

dµk(z) =

1 4λkgk

2

L2_(Ω_k₎,

which ends the proof.

Theorem 3.4. Let s, λ >0. For everyg∈L2_(Ω

k), we have

Tk,`,mFλ,g∗ (y, a) =

Z

Rd

m(az)Ek(iy, z)

1 +λes|z|2

Z ∞

0

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

db b

dµk(z).

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

Tk,`,mFλ,g∗ (y, a) =F

−1

k

_m₍_az₎

1 +λes|z|2

Z ∞

0

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

db b

(y).

The function z→ m(az)

1+λes|z|2

Z ∞

0

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

db b

belongs toL1₍_µ

k). Then

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

Acknowledgments

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