Estimate of K-functionals and modulus of smoothness constructed by generalized spherical mean operator

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: [email protected] MS received 17 January 2013

Abstract. Using a generalized spherical mean operator, we define generalized modu- lus of smoothness in the space L²_k(R^d). 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 onR^d.

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(R^d, w_k). 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 R^d,W the corresponding reflection group,R₊ a positive subsystem ofRandka non-negative andW-invariant function defined onR. The Dunkl operator is defined forf ∈C¹(R^d)by

Djf (x)= ∂f

∂x_j(x)+

α∈R₊

k(α)α_jf (x)−f (σ_α(x))

α, x , x ∈R^d.

235

Here,is the usual Euclidean scalar product onR^dwith the associated norm|.|andσα

the reflection with respect to the hyperplaneH_α orthogonal toα. We consider the weight function

w_k(x)=

α∈R₊

|α, x|^2k(α),

wherew_kis W-invariant and homogeneous of degree 2γwhere

γ =

α∈R₊

k(α).

We letηbe the normalized surface measure on the unit sphereS^d−1inR^dand set dηk(y)=w_k(y)dη(y).

Thenη_kis aW-invariant measure onS^d−1, and we letd_k =η_k(S^d−1).

The Dunkl kernelE_k onR^d×R^d has been introduced by Dunkl in [5]. Fory ∈R^d, the functionx −→E_k(x, y)can be viewed as the solution onR^dof the following initial problem:

Dju(x, y)=y_iu(x, y), 1≤j ≤d, u(0, y)=1.

This kernel has a unique holomorphic extension toC^d×C^d.

Rösler has proved in [12] the following integral representation for the Dunkl kernel, E_k(x, z)=

R^de^y,zdμx(y), x∈R^d, z∈C^d,

whereμxis a probability measure onR^dwith support in the closed ballB(0,|x|)of center 0 and radius|x|.

PROPOSITION 1.1

Letz, w∈C^dandλ∈C. Then (1) E_k(z,0)=1,

(2) E_k(z, w)=E_k(w, z), (3) E_k(λz, w)=E_k(z, λw),

(4) For allν=(ν₁, ..., ν_d)∈N^d, x∈R^d, z∈C^d, we have

|D^ν_zE_k(x, z)| ≤ |x|^|ν|exp(|x||Re(z)|), where

D_z^ν = ∂^|ν|

∂z^ν₁¹. . . ∂z^ν_d^d, |ν| =ν₁+ · · · +ν_d. In particular,

|D^ν_zE_k(ix, z)| ≤ |x|^ν for allx, z∈R^d.

Proof. See [3].

The Dunkl transform is defined forf ∈L¹_k(R^d)=L¹(R^d, w_k(x)dx)by f (ξ )ˆ =c⁻_k¹

R^df (x)E_k(−iξ, x)w_k(x)dx, where the constantc_k is given by

c_k=

R^d e^−|z|

2

2 w_k(z)dz.

The inverse Dunkl transform is defined by the formula f (x)=

R^d

f (ξ )Eˆ _k(ix, ξ )w_k(ξ )dξ, x∈R^d.

The Dunkl LaplacianD_kis defined by Dk =

d

i=1

D_i².

From [11], we have that iff ∈L²_k(R^d),

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 inL²_k(R^d). As in the classical case, a generalized translation operator is defined in the Dunkl (see [13,14]). Namely, for f ∈ L²_k(R^d)and x ∈ R^d we define τx(f ) to be the unique function inL²_k(R^d) satisfying

τ_xf (y)=E_k(ix, y)f (y)ˆ a.e. y∈R^d. Form to Parseval theorem and Proposition 1.1, we see that

τ_xf _L2

k(R^d)≤ f _L2

k(R^d) for allx ∈R^d.

The generalized spherical mean value off ∈L²_k(R^d)is defined by M_hf (x)= 1

d_k

S^d−1τ_x(f )(hy)dηk(y), (x∈R^d, h >0).

We have

M_hf _L2

k(R^d)≤ f _L2

k(R^d). (2)

PROPOSITION 1.2

Letf ∈L²_k(R^d)and fixh >0. Then Mhf ∈L²_k(R^d)and Mhf (ξ )=j_γ+d

2−1(h|ξ|)f (ξ ),ˆ ξ ∈R^d. (3)

Proof. See [8].

Let the functionf (x)∈L²_k(R^d). We define differences of the order m(m∈ 1,2, . . .) with a steph >0.

^m_hf (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

w_m(f, δ)_2,k = sup

0<h≤δ

^m_hf _L2

k(R^d), δ >0.

LetW_2,k^m be the Sobolev space constructed by the operatorD_k, i.e., W^m_2,k= {f ∈L²_k(R^d): D_k^jf ∈L²_k(R^d); j =1,2, . . . , m}. Let us define theK-functional constructed by the spacesL²_k(R^d)andW_2,k^m,

K_m(f, t )_2,k = K(f, t;L²_k(R^d);W_2,k^m)

= inf{ f −g _L2

k(R^d)+t D_k^mg _L2

k(R^d); g∈W_2,k^m}, wheref ∈L²_k(R^d), t >0.

Forα > ⁻₂¹, letj_α(x)be a normalized Bessel function of the first kind, i.e., j_α(x)= 2^α(α+1)Jα(x)

x^α ,

whereJ_α(x)is a Bessel function of the first kind (Chap. 7 of [1]).

The functionj_α(x)is infinitely differentiable,j_α(0)=1.

We understand a generalized exponential function as the function [2]

e_α(x)=j_α(x)+ic_αxj_α+1(x), (4)

wherec_α =(2α+2)⁻¹, i=√

−1.

From (4), we have|1−j_α(x)| ≤ |1−e_α(x)|

2. Main results

Lemma 2.1. Letf (x)∈L²_k(R^d). Then ^m_hf _L2

k(R^d)≤2^m f _L2 k(R^d).

Proof. We use the proof of recurrence formand the formula (2).

Lemma 2.2. Forx ∈R, the following inequalities are fulfilled:

(1) |e_α(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) |j_α(x)| ≤1,

(2) |1−j_α(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 spaceL²_k(R^d);c, c₁, c₂, c₃, . . .are positive constants.

Lemma 2.4. Letf ∈W_2,k^m,t >0. Then wm(f, t )2,k≤c2t^2m D^m_kf _L2

k(R^d).

Proof. Assume thath ∈ (0, t],^m_hf = (I −M_h)^mf is the difference with the steph.

From Proposition 1.2, formula (1) and the Parseval equality, ^m_hf _L2

k(R^d) = (1−j_γ+d

2−1(h|ξ|))^mf (ξ )ˆ _L2 k(R^d); D^m_kf _L2

k(R^d)= |ξ|^2m ˆf (ξ ) _L2

k(R^d). (5)

Formula (5) implies the equality ^m_hf _L2

k(R^d)=h^2m

(1−j_γ+d

2−1(h|ξ|))^m

h^2m|ξ|^2m |ξ|^2mf (ξ )ˆ _L2 k(R^d).

Then

^m_hf _L2

k(R^d)≤h^2m

(1−e_γ+d

2−1(h|ξ|))^2m

(h|ξ|)^2m |ξ|^2mf (ξ )ˆ _L2

k(R^d). (6)

According to Lemma 2.2, for alls∈Rwe have the inequality|(1−eα(x))^2ms^−2m| ≤ c2, wherec2=2^2m. We have

^m_hf _L2

k(R^d) ≤ c₂h^2m |ξ|^2mf (ξ )ˆ _L2 k(R^d)

= c₂h^2m D_k^mf _L2

k(R^d)≤c₂t^2m D^m_kf _L2 k(R^d). Calculating the supremum with respect to allh∈(0, t], we obtain

wm(f, t )2,k≤c2t^2m D^m_kf _L2 k(R^d)

For anyf ∈L²_k(R^d)and any numberν >0, let us define the function P_ν(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 classesW_2,k^m.

Lemma 2.5. For any functionf ∈L²_k(R^d). Then f −Pν(f ) _L2

k(R^d)≤c4 ^m_1/νf _L2

k(R^d), ν >0 Proof. Let|1−j_γ+d

2−1(t )| ≥ c₁ with |t| ≥ 1 (see Lemma 2.3). Using the Parseval equality, we have

f −Pν(f ) _L2

k(R^d) = (1−χν(ξ ))f (ξ )ˆ _L2 k(R^d)

=

1−χ_ν(ξ ) 1−j_γ+d

2−1

|ξ| ν

m 1−j_γ+d 2−1

|ξ| ν

m

× ˆf (ξ ) L²_k(R^d).

Note that sup

|ξ|∈R

1−χ_ν(ξ )

1−j_γ+d 2−1

|ξ|

ν ≤ 1

c^m₁

Then f−P_ν(f ) _L2

k(R^d)≤c⁻₁^m

1−j_γ+d 2−1

|ξ| ν

m

f (ξ )ˆ _L2

k(R^d)=c₄ ^m_1/νf _L2 k(R^d). COROLLARY 2.6

f −Pν(f ) _L2

k(R^d)≤c4wm(f,1/ν)2,k. Lemma 2.7. The following inequality is true:

D^m_k(P_ν(f )) _L2

k(R^d)≤c₅ν^{2m m}_1/νf _L2

k(R^d), ν >0, m∈ {1,2, . . .}. Proof. Using the Parseval equality, we have

D^m_k(P_ν(f )) _L2

k(R^d) = D^m_k(P_ν(f )) _L2

k(R^d)= |ξ|^2mχ_ν(ξ )f (ξ )ˆ _L2 k(R^d)

=

|ξ|^2mχ_ν(ξ )

1−j_γ+d 2−1

|ξ| ν

m 1−j_γ+d 2−1

|ξ| ν

m

× ˆf (ξ ) L²_k(R^d)

.

Note that sup

|ξ|∈R

|ξ|^2mχ_ν(ξ )

|1−j_γ+d 2−1

|ξ| ν

|^m = ν^2m sup

|ξ|≤ν

|ξ| ν

2m

|1−j_γ+d 2−1

|ξ| ν

|^m

= ν^2msup

|t|≤1

t^2m

|1−j_γ+d

2−1(t )|^m.

Let

c₅= sup

|t|≤1

t^2m

|1−j_γ+d

2−1(t )|^m. Then, we have

D^m_k(Pν(f )) _L2

k(R^d)≤c5ν^{2m m}_1/νf _L2 k. COROLLARY 2.8

D_k^m(P_ν(f )) _L2

k(R^d) ≤c5ν^2mw_m(f,1/ν)2,k.

Theorem 2.9. One can find positive numbersc₆andc₇which the inequality c₆w_m(f, δ)_2,k ≤K_m(f, δ^2m)_2,k≤c₇w_m(f, δ)_2,k,

f ∈L²_k(R^d), δ >0.

Proof. Firstly prove of the inequality c₆w_m(f, δ)_2,k ≤K_m(f, δ^2m)_2,k.

Leth∈(0, δ], g∈W_2,k^m. Using Lemmas 2.1 and 2.4, we have ^m_hf _L2

k(R^d) ≤ ^m_h(f−g) _L2

k(R^d)+ ^m_hg _L2 k(R^d)

≤ 2^m f −g _L2

k(R^d)+c2h^2m D_k^mg _L2 k(R^d)

≤ c₈( f −g _L2

k(R^d)+δ^2m D_k^mg _L2 k(R^d)),

wherec8 = max(2^m, c2). Calculating the supremum with respect toh ∈ (0, δ]and the infimum with respect to all possible functionsg∈W^m_2,k, we obtain

w_m(f, δ)_2,k ≤c₈K_m(f, δ^2m)_2,k, whence we get the inequality.

Now, we prove the inequality

Km(f, δ^2m)2,k ≤c7wm(f, δ)2,k.

SincePν(f )∈W_2,k^m, by the definition of a K-functional we have Km(f, δ^2m)2,k ≤ f −Pν(f ) _L2

k(R^d)+δ^2m D_k^mPν(f ) _L2 k(R^d). Using Corollaries 2.6 and 2.8, we obtain

K_m(f, δ^2m)2,k ≤c4w_m(f,1/ν)2,k+c5ν^2mδ^2mw_m(f,1/ν)2,k, K_m(f, δ^2m)_2,k ≤c₄w_m(f,1/ν)2,k+c₅(νδ)^2mw_m(f,1/ν)2,k. Sinceνis an arbitrary positive value, choosingν=1/δ, we obtain the inequality.

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