Estimate of K-functionals and modulus of smoothness constructed by generalized spherical mean operator
M EL HAMMA and R DAHER
Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca, Morocco
E-mail: m_elhamma@yahoo.fr MS received 17 January 2013
Abstract. Using a generalized spherical mean operator, we define generalized modu- lus of smoothness in the space L2k(Rd). Based on the Dunkl operator we define Sobolev-type space and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the Dunkl transform onRd.
Keywords. Dunkl operator; generalized spherical mean operator; K-functional;
modulus of smoothness.
AMS Subject Classification. 47B48, 33C52, 33C67.
1. Introduction and preliminaries
In [2], Belkina and Platonov proved the equivalence theorem for a K-functional and a modulus of smoothness for the Dunkl transform in the Hilbert space L2(R,|x|2α+1), α >−1/2, using a Dunkl translation operator.
In this paper, we prove the analog of this result (see [2]) in the Hilbert space L2(Rd, wk). For this purpose, we use a generalized spherical mean operator in the place of the Dunkl translation operator.
Dunkl [4] defined a family of first-order differential-difference operators related to some reflection groups. These operators generalize in a certain manner the usual differ- entiation and have gained considerable interest in various fields of mathematics and also in physical applications. The theory of Dunkl operators provides generalizations of vari- ous multivariable analytic structures. Among others, we cite the exponential function, the Fourier transform and the translation operator. For more details about these operators, see [3,4,6,7,9,10,12] and the references therein.
LetR be a root system in Rd,W the corresponding reflection group,R+ a positive subsystem ofRandka non-negative andW-invariant function defined onR. The Dunkl operator is defined forf ∈C1(Rd)by
Djf (x)= ∂f
∂xj(x)+
α∈R+
k(α)αjf (x)−f (σα(x))
α, x , x ∈Rd.
235
Here,is the usual Euclidean scalar product onRdwith the associated norm|.|andσα
the reflection with respect to the hyperplaneHα orthogonal toα. We consider the weight function
wk(x)=
α∈R+
|α, x|2k(α),
wherewkis W-invariant and homogeneous of degree 2γwhere
γ =
α∈R+
k(α).
We letηbe the normalized surface measure on the unit sphereSd−1inRdand set dηk(y)=wk(y)dη(y).
Thenηkis aW-invariant measure onSd−1, and we letdk =ηk(Sd−1).
The Dunkl kernelEk onRd×Rd has been introduced by Dunkl in [5]. Fory ∈Rd, the functionx −→Ek(x, y)can be viewed as the solution onRdof the following initial problem:
Dju(x, y)=yiu(x, y), 1≤j ≤d, u(0, y)=1.
This kernel has a unique holomorphic extension toCd×Cd.
Rösler has proved in [12] the following integral representation for the Dunkl kernel, Ek(x, z)=
Rdey,zdμx(y), x∈Rd, z∈Cd,
whereμxis a probability measure onRdwith support in the closed ballB(0,|x|)of center 0 and radius|x|.
PROPOSITION 1.1
Letz, w∈Cdandλ∈C. Then (1) Ek(z,0)=1,
(2) Ek(z, w)=Ek(w, z), (3) Ek(λz, w)=Ek(z, λw),
(4) For allν=(ν1, ..., νd)∈Nd, x∈Rd, z∈Cd, we have
|DνzEk(x, z)| ≤ |x||ν|exp(|x||Re(z)|), where
Dzν = ∂|ν|
∂zν11. . . ∂zνdd, |ν| =ν1+ · · · +νd. In particular,
|DνzEk(ix, z)| ≤ |x|ν for allx, z∈Rd.
Proof. See [3].
The Dunkl transform is defined forf ∈L1k(Rd)=L1(Rd, wk(x)dx)by f (ξ )ˆ =c−k1
Rdf (x)Ek(−iξ, x)wk(x)dx, where the constantck is given by
ck=
Rd e−|z|
2
2 wk(z)dz.
The inverse Dunkl transform is defined by the formula f (x)=
Rd
f (ξ )Eˆ k(ix, ξ )wk(ξ )dξ, x∈Rd.
The Dunkl LaplacianDkis defined by Dk =
d
i=1
Di2.
From [11], we have that iff ∈L2k(Rd),
Dkf (ξ )= −|ξ|2f (ξ ).ˆ (1)
The Dunkl transform shares several properties with its counterpart in the classical case.
We mention here, in particular that Parseval theorem holds inL2k(Rd). As in the classical case, a generalized translation operator is defined in the Dunkl (see [13,14]). Namely, for f ∈ L2k(Rd)and x ∈ Rd we define τx(f ) to be the unique function inL2k(Rd) satisfying
τxf (y)=Ek(ix, y)f (y)ˆ a.e. y∈Rd. Form to Parseval theorem and Proposition 1.1, we see that
τxf L2
k(Rd)≤ f L2
k(Rd) for allx ∈Rd.
The generalized spherical mean value off ∈L2k(Rd)is defined by Mhf (x)= 1
dk
Sd−1τx(f )(hy)dηk(y), (x∈Rd, h >0).
We have
Mhf L2
k(Rd)≤ f L2
k(Rd). (2)
PROPOSITION 1.2
Letf ∈L2k(Rd)and fixh >0. Then Mhf ∈L2k(Rd)and Mhf (ξ )=jγ+d
2−1(h|ξ|)f (ξ ),ˆ ξ ∈Rd. (3)
Proof. See [8].
Let the functionf (x)∈L2k(Rd). We define differences of the order m(m∈ 1,2, . . .) with a steph >0.
mhf (x)=(I−Mh)mf (x), whereI is the unit operator.
For any positive integer m, we define the generalized module of smoothness of themth order by the formula
wm(f, δ)2,k = sup
0<h≤δ
mhf L2
k(Rd), δ >0.
LetW2,km be the Sobolev space constructed by the operatorDk, i.e., Wm2,k= {f ∈L2k(Rd): Dkjf ∈L2k(Rd); j =1,2, . . . , m}. Let us define theK-functional constructed by the spacesL2k(Rd)andW2,km,
Km(f, t )2,k = K(f, t;L2k(Rd);W2,km)
= inf{ f −g L2
k(Rd)+t Dkmg L2
k(Rd); g∈W2,km}, wheref ∈L2k(Rd), t >0.
Forα > −21, letjα(x)be a normalized Bessel function of the first kind, i.e., jα(x)= 2α(α+1)Jα(x)
xα ,
whereJα(x)is a Bessel function of the first kind (Chap. 7 of [1]).
The functionjα(x)is infinitely differentiable,jα(0)=1.
We understand a generalized exponential function as the function [2]
eα(x)=jα(x)+icαxjα+1(x), (4)
wherecα =(2α+2)−1, i=√
−1.
From (4), we have|1−jα(x)| ≤ |1−eα(x)|
2. Main results
Lemma 2.1. Letf (x)∈L2k(Rd). Then mhf L2
k(Rd)≤2m f L2 k(Rd).
Proof. We use the proof of recurrence formand the formula (2).
Lemma 2.2. Forx ∈R, the following inequalities are fulfilled:
(1) |eα(x)| ≤1, (2) |1−eα(x)| ≤2|x|,
(3) |1−eα(x)| ≥cwith|x| ≥1, wherec >0 is a certain constant which depends only onα.
Proof. See [2].
Lemma 2.3. Forx ∈R, the following inequalities are fulfilled:
(1) |jα(x)| ≤1,
(2) |1−jα(x)| ≥c1with|x| ≥1, wherec1>0 is a certain constant which depends only onα.
Proof. Analog of proof of Lemma 2.2.
In what follows,f (x)is an arbitrary function of the spaceL2k(Rd);c, c1, c2, c3, . . .are positive constants.
Lemma 2.4. Letf ∈W2,km,t >0. Then wm(f, t )2,k≤c2t2m Dmkf L2
k(Rd).
Proof. Assume thath ∈ (0, t],mhf = (I −Mh)mf is the difference with the steph.
From Proposition 1.2, formula (1) and the Parseval equality, mhf L2
k(Rd) = (1−jγ+d
2−1(h|ξ|))mf (ξ )ˆ L2 k(Rd); Dmkf L2
k(Rd)= |ξ|2m ˆf (ξ ) L2
k(Rd). (5)
Formula (5) implies the equality mhf L2
k(Rd)=h2m
(1−jγ+d
2−1(h|ξ|))m
h2m|ξ|2m |ξ|2mf (ξ )ˆ L2 k(Rd).
Then
mhf L2
k(Rd)≤h2m
(1−eγ+d
2−1(h|ξ|))2m
(h|ξ|)2m |ξ|2mf (ξ )ˆ L2
k(Rd). (6)
According to Lemma 2.2, for alls∈Rwe have the inequality|(1−eα(x))2ms−2m| ≤ c2, wherec2=22m. We have
mhf L2
k(Rd) ≤ c2h2m |ξ|2mf (ξ )ˆ L2 k(Rd)
= c2h2m Dkmf L2
k(Rd)≤c2t2m Dmkf L2 k(Rd). Calculating the supremum with respect to allh∈(0, t], we obtain
wm(f, t )2,k≤c2t2m Dmkf L2 k(Rd)
For anyf ∈L2k(Rd)and any numberν >0, let us define the function Pν(f )(x)=−1(f (ξ )χˆ ν(ξ )),
whereχν(ξ )is the function defined byχν(ξ )=1, for|ξ| ≤νandχ (ξ )=0, for|ξ|> ν, −1 is the inverse Dunkl transform. One can easily prove that the function Pν(f )is infinitely differentiable and belongs to all classesW2,km.
Lemma 2.5. For any functionf ∈L2k(Rd). Then f −Pν(f ) L2
k(Rd)≤c4 m1/νf L2
k(Rd), ν >0 Proof. Let|1−jγ+d
2−1(t )| ≥ c1 with |t| ≥ 1 (see Lemma 2.3). Using the Parseval equality, we have
f −Pν(f ) L2
k(Rd) = (1−χν(ξ ))f (ξ )ˆ L2 k(Rd)
=
1−χν(ξ ) 1−jγ+d
2−1
|ξ| ν
m 1−jγ+d 2−1
|ξ| ν
m
× ˆf (ξ ) L2k(Rd).
Note that sup
|ξ|∈R
1−χν(ξ )
1−jγ+d 2−1
|ξ|
ν ≤ 1
cm1
Then f−Pν(f ) L2
k(Rd)≤c−1m
1−jγ+d 2−1
|ξ| ν
m
f (ξ )ˆ L2
k(Rd)=c4 m1/νf L2 k(Rd). COROLLARY 2.6
f −Pν(f ) L2
k(Rd)≤c4wm(f,1/ν)2,k. Lemma 2.7. The following inequality is true:
Dmk(Pν(f )) L2
k(Rd)≤c5ν2m m1/νf L2
k(Rd), ν >0, m∈ {1,2, . . .}. Proof. Using the Parseval equality, we have
Dmk(Pν(f )) L2
k(Rd) = Dmk(Pν(f )) L2
k(Rd)= |ξ|2mχν(ξ )f (ξ )ˆ L2 k(Rd)
=
|ξ|2mχν(ξ )
1−jγ+d 2−1
|ξ| ν
m 1−jγ+d 2−1
|ξ| ν
m
× ˆf (ξ ) L2k(Rd)
.
Note that sup
|ξ|∈R
|ξ|2mχν(ξ )
|1−jγ+d 2−1
|ξ| ν
|m = ν2m sup
|ξ|≤ν
|ξ| ν
2m
|1−jγ+d 2−1
|ξ| ν
|m
= ν2msup
|t|≤1
t2m
|1−jγ+d
2−1(t )|m.
Let
c5= sup
|t|≤1
t2m
|1−jγ+d
2−1(t )|m. Then, we have
Dmk(Pν(f )) L2
k(Rd)≤c5ν2m m1/νf L2 k. COROLLARY 2.8
Dkm(Pν(f )) L2
k(Rd) ≤c5ν2mwm(f,1/ν)2,k.
Theorem 2.9. One can find positive numbersc6andc7which the inequality c6wm(f, δ)2,k ≤Km(f, δ2m)2,k≤c7wm(f, δ)2,k,
f ∈L2k(Rd), δ >0.
Proof. Firstly prove of the inequality c6wm(f, δ)2,k ≤Km(f, δ2m)2,k.
Leth∈(0, δ], g∈W2,km. Using Lemmas 2.1 and 2.4, we have mhf L2
k(Rd) ≤ mh(f−g) L2
k(Rd)+ mhg L2 k(Rd)
≤ 2m f −g L2
k(Rd)+c2h2m Dkmg L2 k(Rd)
≤ c8( f −g L2
k(Rd)+δ2m Dkmg L2 k(Rd)),
wherec8 = max(2m, c2). Calculating the supremum with respect toh ∈ (0, δ]and the infimum with respect to all possible functionsg∈Wm2,k, we obtain
wm(f, δ)2,k ≤c8Km(f, δ2m)2,k, whence we get the inequality.
Now, we prove the inequality
Km(f, δ2m)2,k ≤c7wm(f, δ)2,k.
SincePν(f )∈W2,km, by the definition of a K-functional we have Km(f, δ2m)2,k ≤ f −Pν(f ) L2
k(Rd)+δ2m DkmPν(f ) L2 k(Rd). Using Corollaries 2.6 and 2.8, we obtain
Km(f, δ2m)2,k ≤c4wm(f,1/ν)2,k+c5ν2mδ2mwm(f,1/ν)2,k, Km(f, δ2m)2,k ≤c4wm(f,1/ν)2,k+c5(νδ)2mwm(f,1/ν)2,k. Sinceνis an arbitrary positive value, choosingν=1/δ, we obtain the inequality.
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