Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 235–242.
c Indian Academy of Sciences
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 modulus 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 on Rd .
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 differentiation and have gained considerable interest in various fields of mathematics and also
in physical applications. The theory of Dunkl operators provides generalizations of various 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.
Let R be a root system in Rd , W the corresponding reflection group, R+ a positive
subsystem of R and k a non-negative and W -invariant function defined on R. The Dunkl
operator is defined for f ∈ C 1 (Rd ) by
Dj f (x) =
f (x) − f (σα (x))
∂f
,
(x) +
k(α)αj
∂xj
α, x
x ∈ Rd .
α∈R+
235
236
M El Hamma and R Daher
Here , is the usual Euclidean scalar product on Rd with the associated norm |.| and σα
the reflection with respect to the hyperplane Hα orthogonal to α. We consider the weight
function
|α, x|2k(α) ,
wk (x) =
α∈R+
where wk is W-invariant and homogeneous of degree 2γ where
γ =
k(α).
α∈R+
We let η be the normalized surface measure on the unit sphere Sd−1 in Rd and set
dηk (y) = wk (y)dη(y).
Then ηk is a W -invariant measure on Sd−1 , and we let dk = ηk (Sd−1 ).
The Dunkl kernel Ek on Rd × Rd has been introduced by Dunkl in [5]. For y ∈ Rd ,
the function x −→ Ek (x, y) can be viewed as the solution on Rd of the following initial
problem:
Dj u(x, y) = yi u(x, y),
1 ≤ j ≤ d,
u(0, y) = 1.
This kernel has a unique holomorphic extension to Cd × Cd .
Rösler has proved in [12] the following integral representation for the Dunkl kernel,
ey,z dμx (y), x ∈ Rd , z ∈ Cd ,
Ek (x, z) =
Rd
where μx is a probability measure on Rd with support in the closed ball B(0, |x|) of center
0 and radius |x|.
PROPOSITION 1.1
Let z, w ∈ Cd and λ ∈ C. Then
(1)
(2)
(3)
(4)
Ek (z, 0) = 1,
Ek (z, w) = Ek (w, z),
Ek (λz, w) = Ek (z, λw),
For all ν = (ν1 , ..., νd ) ∈ Nd , x ∈ Rd , z ∈ Cd , we have
|Dzν Ek (x, z)| ≤ |x||ν| exp(|x||Re(z)|),
where
Dzν =
∂ |ν|
,
∂z1ν1 . . . ∂zdνd
In particular,
|Dzν Ek (ix, z)| ≤ |x|ν
for all x, z ∈ Rd .
|ν| = ν1 + · · · + νd .
Estimate of K-functionals and modulus of smoothness
237
Proof. See [3].
The Dunkl transform is defined for f ∈ L1k (Rd ) = L1 (Rd , wk (x)dx) by
fˆ(ξ ) = ck−1
f (x)Ek (−iξ, x)wk (x)dx,
Rd
where the constant ck is given by
−|z|2
e 2 wk (z)dz.
ck =
Rd
The inverse Dunkl transform is defined by the formula
fˆ(ξ )Ek (ix, ξ )wk (ξ )dξ, x ∈ Rd .
f (x) =
Rd
The Dunkl Laplacian Dk is defined by
Dk =
d
Di2 .
i=1
From [11], we have that if f ∈ L2k (Rd ),
2
D
k f (ξ ) = −|ξ | fˆ(ξ ).
(1)
The Dunkl transform shares several properties with its counterpart in the classical case.
We mention here, in particular that Parseval theorem holds in L2k (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 in L2k (Rd )
satisfying
τ
x f (y) = Ek (ix, y)fˆ(y)
a.e. y ∈ Rd .
Form to Parseval theorem and Proposition 1.1, we see that
τx f L2 (Rd ) ≤ f L2 (Rd )
k
k
for all x ∈ Rd .
The generalized spherical mean value of f ∈ L2k (Rd ) is defined by
1
τx (f )(hy)dηk (y), (x ∈ Rd , h > 0).
Mh f (x) =
dk Sd−1
We have
Mh f L2 (Rd ) ≤ f L2 (Rd ) .
k
(2)
k
PROPOSITION 1.2
Let f ∈ L2k (Rd ) and fix h > 0. Then Mh f ∈ L2k (Rd ) and
M
h f (ξ ) = jγ + d −1 (h|ξ |)fˆ(ξ ),
2
ξ ∈ Rd .
(3)
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M El Hamma and R Daher
Proof. See [8].
Let the function f (x) ∈ L2k (Rd ). We define differences of the order m (m ∈ 1, 2, . . .)
with a step h > 0.
m
m
h f (x) = (I − Mh ) f (x),
where I is the unit operator.
For any positive integer m, we define the generalized module of smoothness of the mth
order by the formula
wm (f, δ)2,k = sup m
h f L2 (Rd ) ,
k
0<h≤δ
δ > 0.
m be the Sobolev space constructed by the operator D , i.e.,
Let W2,k
k
j
2
d
2
d
Wm
2,k = {f ∈ Lk (R ) : Dk f ∈ Lk (R ); j = 1, 2, . . . , m}.
m ,
Let us define the K-functional constructed by the spaces L2k (Rd ) and W2,k
m
)
Km (f, t)2,k = K(f, t; L2k (Rd ); W2,k
m
},
= inf{f − gL2 (Rd ) + tDkm gL2 (Rd ) ; g ∈ W2,k
k
k
where f ∈ L2k (Rd ), t > 0.
For α > −1
2 , let jα (x) be a normalized Bessel function of the first kind, i.e.,
jα (x) =
2α (α + 1)Jα (x)
,
xα
where Jα (x) is a Bessel function of the first kind (Chap. 7 of [1]).
The function jα (x) is infinitely differentiable, jα (0) = 1.
We understand a generalized exponential function as the function [2]
eα (x) = jα (x) + icα xjα+1 (x),
√
where cα = (2α + 2)−1 , i = −1.
From (4), we have |1 − jα (x)| ≤ |1 − eα (x)|
(4)
2. Main results
Lemma 2.1. Let f (x) ∈ L2k (Rd ). Then
m
m
h f L2 (Rd ) ≤ 2 f L2 (Rd ).
k
k
Proof. We use the proof of recurrence for m and the formula (2).
Lemma 2.2. For x ∈ R, the following inequalities are fulfilled:
(1) |eα (x)| ≤ 1,
(2) |1 − eα (x)| ≤ 2|x|,
(3) |1 − eα (x)| ≥ c with |x| ≥ 1, where c > 0 is a certain constant which depends only
on α.
Estimate of K-functionals and modulus of smoothness
239
Proof. See [2].
Lemma 2.3. For x ∈ R, the following inequalities are fulfilled:
(1) |jα (x)| ≤ 1,
(2) |1 − jα (x)| ≥ c1 with |x| ≥ 1, where c1 > 0 is a certain constant which depends only
on α.
Proof. Analog of proof of Lemma 2.2.
In what follows, f (x) is an arbitrary function of the space L2k (Rd ); c, c1 , c2 , c3 , . . . are
positive constants.
m , t > 0. Then
Lemma 2.4. Let f ∈ W2,k
wm (f, t)2,k ≤ c2 t 2m Dm
k f L2 (Rd ) .
k
m
Proof. Assume that h ∈ (0, t], m
h f = (I − Mh ) f is the difference with the step h.
From Proposition 1.2, formula (1) and the Parseval equality,
m ˆ
m
h f L2 (Rd ) = (1 − jγ + d −1 (h|ξ |)) f (ξ )L2 (Rd ) ;
k
k
2
Dkm f L2 (Rd ) = |ξ |2m fˆ(ξ )L2 (Rd ).
k
(5)
k
Formula (5) implies the equality
m
h f L2k (Rd )
=h
m
h f L2k (Rd )
≤h
2m
(1 − jγ + d −1 (h|ξ |))m
2
h2m |ξ |2m
|ξ |2m fˆ(ξ )L2 (Rd ).
k
Then
2m
(1 − eγ + d −1 (h|ξ |))2m
2
(h|ξ |)2m
|ξ |2m fˆ(ξ )L2 (Rd ).
k
(6)
According to Lemma 2.2, for all s ∈ R we have the inequality |(1 − eα (x))2m s −2m | ≤
c2 , where c2 = 22m . We have
2m
2m ˆ
m
h f L2 (Rd ) ≤ c2 h |ξ | f (ξ )L2 (Rd )
k
k
= c2 h2m Dkm f L2 (Rd ) ≤ c2 t 2m Dkm f L2 (Rd ) .
k
k
Calculating the supremum with respect to all h ∈ (0, t], we obtain
wm (f, t)2,k ≤ c2 t 2m Dm
k f L2 (Rd )
k
For any f ∈
L2k (Rd )
and any number ν > 0, let us define the function
Pν (f )(x) = −1 (fˆ(ξ )χν (ξ )),
where χν (ξ ) is the function defined by χν (ξ ) = 1, for |ξ | ≤ ν and χ (ξ ) = 0, for |ξ | > ν,
−1 is the inverse Dunkl transform. One can easily prove that the function Pν (f ) is
m .
infinitely differentiable and belongs to all classes W2,k
240
M El Hamma and R Daher
Lemma 2.5. For any function f ∈ L2k (Rd ). Then
f − Pν (f )L2 (Rd ) ≤ c4 m
1/ν f L2 (Rd ) ,
k
ν>0
k
Proof. Let |1 − jγ + d −1 (t)| ≥ c1 with |t| ≥ 1 (see Lemma 2.3). Using the Parseval
2
equality, we have
f − Pν (f )L2 (Rd ) = (1 − χν (ξ ))fˆ(ξ )L2 (Rd )
k
k
m
1 − χν (ξ )
|ξ |
= m 1 − jγ + d −1
2
|ξ |
ν
1−jγ + d −1 ν
2
×fˆ(ξ )
L2k (Rd ).
Note that
1
1 − χν (ξ )
≤ m
sup |ξ
|
c
|ξ |∈R 1 − j
1
γ + d −1 ν
2
m
Then f −Pν (f )L2 (Rd ) ≤ c1−m 1−jγ + d −1 |ξν |
fˆ(ξ )L2 (Rd )=c4 m
1/ν f L2 (Rd ) .
k
k
2
k
COROLLARY 2.6
f − Pν (f )L2 (Rd ) ≤ c4 wm (f, 1/ν)2,k .
k
Lemma 2.7. The following inequality is true:
Dkm (Pν (f ))L2 (Rd ) ≤ c5 ν 2m m
1/ν f L2 (Rd ) ,
k
k
ν > 0, m ∈ {1, 2, . . .}.
Proof. Using the Parseval equality, we have
2m
m
ˆ
Dkm (Pν (f ))L2 (Rd ) = D
k (Pν (f ))L2 (Rd ) = |ξ | χν (ξ )f (ξ )L2 (Rd )
k
k
k
m
|ξ |2m χν (ξ )
|ξ |
1
−
j
= d
m
γ + 2 −1
|ξ |
ν
1 − jγ + d −1 ν
ˆ
×f (ξ )
2
.
L2k (Rd )
Note that
sup
|ξ |∈R
|ξ |2m χ
ν (ξ )
|1 − jγ + d −1
2
|ξ |
ν
|m
= ν 2m sup
|ξ |≤ν
|ξ |
ν
|1 − jγ + d −1
2
= ν 2m sup
|t|≤1
2m
|ξ |
ν
t 2m
|1 − jγ + d −1 (t)|m
2
.
|m
Estimate of K-functionals and modulus of smoothness
241
Let
c5 = sup
|t|≤1
t 2m
|1 − jγ + d −1 (t)|m
.
2
Then, we have
Dkm (Pν (f ))L2 (Rd ) ≤ c5 ν 2m m
1/ν f L2 .
k
k
COROLLARY 2.8
Dkm (Pν (f ))L2 (Rd ) ≤ c5 ν 2m wm (f, 1/ν)2,k .
k
Theorem 2.9. One can find positive numbers c6 and c7 which the inequality
c6 wm (f, δ)2,k ≤ Km (f, δ 2m )2,k ≤ c7 wm (f, δ)2,k ,
f ∈ L2k (Rd ), δ > 0.
Proof. Firstly prove of the inequality
c6 wm (f, δ)2,k ≤ Km (f, δ 2m )2,k .
m . Using Lemmas 2.1 and 2.4, we have
Let h ∈ (0, δ], g ∈ W2,k
m
m
m
h f L2 (Rd ) ≤ h (f − g)L2 (Rd ) + h gL2 (Rd )
k
k
k
≤ 2m f − gL2 (Rd ) + c2 h2m Dkm gL2 (Rd )
k
k
≤ c8 (f − gL2 (Rd ) + δ 2m Dkm gL2 (Rd ) ),
k
k
where c8 =
Calculating the supremum with respect to h ∈ (0, δ] and the
infimum with respect to all possible functions g ∈ Wm
2,k , we obtain
max(2m , c2 ).
wm (f, δ)2,k ≤ c8 Km (f, δ 2m )2,k ,
whence we get the inequality.
Now, we prove the inequality
Km (f, δ 2m )2,k ≤ c7 wm (f, δ)2,k .
m , by the definition of a K-functional we have
Since Pν (f ) ∈ W2,k
Km (f, δ 2m )2,k ≤ f − Pν (f )L2 (Rd ) + δ 2m Dkm Pν (f )L2 (Rd ) .
k
k
Using Corollaries 2.6 and 2.8, we obtain
Km (f, δ 2m )2,k ≤ c4 wm (f, 1/ν)2,k + c5 ν 2m δ 2m wm (f, 1/ν)2,k ,
Km (f, δ 2m )2,k ≤ c4 wm (f, 1/ν)2,k + c5 (νδ)2m wm (f, 1/ν)2,k .
Since ν is an arbitrary positive value, choosing ν = 1/δ, we obtain the inequality.
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M El Hamma and R Daher
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