Equivalence of K Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line

http://dx.doi.org/10.4236/apm.2015.56035

Equivalence of

K-

Functionals and Modulus

of Smoothness Generated by a Generalized

Dunkl Operator on the Real Line

Reem Fahad Al Subaie

1

, Mohamed Ali Mourou

2

1_{Department of Mathematics, College of Sciences for Girls, University of Dammam, Dammam, Kingdom of} Saudi Arabia

2_{Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia} Email: [email protected], [email protected]

Received 23 April 2015; accepted 25 May 2015; published 28 May 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

This paper is intended to establish the equivalence between K-functionals and modulus of smooth- ness tied to a Dunkl type operator on the real line.

Keywords

Differential-Difference Operator, Generalized Fourier Transform, Generalized Translation Operators, K-Functionals, Modulus of Smoothness

1. Introduction

Consider the first-order singular differential-difference operator on the real line

( )

( )

1

( )

( )

( )

2 ,

2

f x f x f x

f x f x n

x x

α − − −

 

′

Λ = +_ + _ −

 

where α > −1 2 and n=0,1,. For n=0, we regain the differential-difference operator

( )

( )

1

( )

( )

, 2

f x f x

D f x f x

x

α α

− −

 

′

= +_ + _

 

theory of spherical harmonics. The one-dimensional Dunkl operator Dα plays a major role in the study of quantum harmonic oscillators governed by Wigner’s commutation rules [4]-[6].

The authors have developed in [7] [8] a new harmonic analysis on the real line related to the differential-dif- ference operator Λ in which several classical analytic structures such as the Fourier transform, the translation operators, the convolution product, ..., were generalized. With the help of the translation operators tied to Λ, we construct in this paper generalized modulus of smoothness in the Hilbert space _L2

(

_,x2α+1_dx

)

_{. Next, we}

define Sobolev type spaces and K-functionals generated by Λ. Using essentially the properties of the Fourier transform associated to Λ, we establish the equivalence between K-functionals and modulus of smoothness.

In the classical theory of approximation of functions on , the modulus of smoothness are basically built by means of the translation operators f → f x

(

+y

)

. As the notion of translation operators was extended to vari-ous contexts (see [9] [10] and the references therein), many generalized modulus of smoothness have been dis-covered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [11]-[13] and references therein).

In addition to modulus of smoothness, the K-functionals introduced by J. Peetre [14] have turned out to be a simple and efficient tool for the description of smoothness properties of functions. The study of the connection between these two quantities is one of the main problems in the theory of approximation of functions. In the classical setting, the equivalence of modulus of smoothness and K-functionals has been established in [15]. For various generalized modulus of smoothness these problems are studied, for example, in [16]-[19]. It is pointed out that the results obtained in [16] emerge as easy consequences of those stated in the present paper by simply taking n=0.

2. Preliminaries

In this section, we develop some results from harmonic analysis related to the differential-difference operator

Λ. Further details can be found in [7] [8]. In all what follows assume α > −1 2 and n a non-negative integer. The one-dimensional Dunkl kernel is defined by

( )

_{( ) ( ) ( ) ( )}

1

2 1

z

e_α z j_α iz j_α iz z

α +

= + ∈

+  (1)

where

( )

(

) ( ) ( )

₍

2

_{) (}

)

0

1 2

1

! 1

n n

n

z

j z z

n n

α α _α

∞

=

−

= Γ + ∈

Γ + +

∑



is the normalized spherical Bessel function of index

α

. It is well-known that the functions eα

( )

λ⋅ , λ ∈, are solutions of the differential-difference equation

( )

, 0 1.

D u_α =λu u = (2) Furthermore, we have the Laplace type integral representations:

( )

1

(

₂

)

1 2

(

)

1 1 1 e d ,

zt

e_α z a_α t α− t t

−

=

_∫

− + (3)

( )

1

(

₂

)

1 2

0

2 1 cos d ,

jα z aα t α zt t

−

=

_∫

− (4) where

(

)

(

)

1 .

π 1 2

aα

α

α

Γ +

=

Γ + (5)

The following properties will be useful for the sequel.

Lemma 1 1) For all x∈, e_α

( )

ix ≤1.

3) For all x∈ −

{ }

0 , eα

( )

ix ≠1. 4) For all x∈,

( )

(

₍

1

)

₎

1 .

π 3 2

e_α ix

α

x x

α

Γ +

− ≤ ≤

Γ +

Proof. Assertions (1) and (2) are proved in [16]. By (1), (4) and the fact that

(

)

1 2

1 ₂

0

1 2aα 1 t α dt −

=

_∫

−

we have

( )

( )

1

(

₂

)

1 2

(

)

0

1−e_α ix ≥ −1 j_α x =2a_α

∫

1−t α− 1 cos− xt d .t

Clearly the integral above is null only for x=0, which proves assertion (3). Let us check assertion (4). Using (3) and the fact that

(

)

1 2

(

)

1 ₂

1

1 a_α 1 t α− 1 t dt −

=

_∫

− + (6) 1 e₋ iz _{≤ z} for all_z∈

we get

( )

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1 2

1 ₂

1

1 2

1 ₂

1

1 2

1 ₂

1

1 1 1 1 e d

1 1 1 e d

1 1 d

ixt

ixt

e ix a t t t

a t t t

a x t t t t

α

α α

α α

α α

− −

− −

− −

− = − + −

≤ − + −

≤ − +

∫

∫

∫

By (6),

(

)

1 2

(

)

(

)

1 2

(

)

1 ₂ 1 ₂

11 1 d 11 1 d

a x_α

∫

₋ −t α− +t t t≤a x_α

∫

₋ −t α− +t t= x

Moreover,

(

)

(

)

(

)

(

₍

)

₎

1 2

1 ₂

1

1 2

1 ₂

0

1 1 d

1

2 1 d ,

1 2 π 3 2

a x t t t t

a x

a x t t t x

α α

α _α

α

α

α α

− −

−

− +

Γ +

= − = =

+ Γ +

∫

∫

which concludes the proof.

Notation 1 Put

(

)

(

)

2

2 2

1

.

2 1

m_α

α+ _α

=

Γ +

We denote by

 2

( )

L_α  the class of measurable functions f on  for which

( )

(

₂

)

1 2

2 1

2, d .

f _α =

∫

_ f x x α+ x < ∞

( )

(

)

( )

,

d

sup 1 .

d n m

m n n

x

q f x f x

x

∈

= + < ∞



The topology of 

( )

 is defined by the semi-norms q_{m n,} , m n, =0,1,.

 _n

( )

_ the subspace of 

( )

 consisting of functions f such that

( )

(2 1)

_{( )}

0 n 0 0.

f == f − =

 ′

( )

_ the space of tempered distributions on .

 _n′

( )

_ the topological dual of _n

( )

 .

Clearly Λ is a linear bounded operator from _n

( )

 into itself. Accordingly, if S∈_n′

( )

 define Λ ∈S _n′

( )



by

( )

, , , _n

Sψ S ψ ψ

Λ = − Λ ∈ 

For k=0,1, and S∈′

( )

 , let k

( )

x S∈′  be defined by

( )

, , , .

k k

x Sψ = S xψ ψ∈ 

Definition 1 The generalized Fourier transform of a function f∈_n

( )

 is defined by

( )( )

( )

(

)

2 2 1

2 d , .

n n

f λ f x eα i x xλ α x λ

+ + +

=

_∫

_ − ∈



Remark 1 If n=0 then  reduces to the Dunkl transform with parameter α +1 2 associated with the reflection group ₂ on  (see [3]).

Theorem 1 The generalized Fourier transform  is a topological isomorphism from _n

( )

 onto 

( )

 . The inverse transform is given by

( )( )

( )

( ) ( )

1 2

2 d ,

n

n

g x x g λ e_α i xλ σ λ

−

+

=

_∫

 _

where

( )

2 4 1

2

dσ λ mα _n λ α n d .λ

+ + +

=

Theorem 2 1) For every f ∈_n

( )

 we have the Plancherel formula

( )

2 2 1

( )( )

2

( )

d d .

f x x α+ x= f λ σ λ

∫



∫



2) The generalized Fourier transform  extends uniquely to an isometric isomorphism from 2

( )

L_α  onto

(

)

2 ,

L σ .

Definition 2 The generalized Fourier transform of a distribution S∈_n′

( )

 is defined by

( )

1

( )

( )

, , , .

S ψ = S − ψ ψ∈ 

  

Theorem 3 The generalized Fourier transform  is one-to-one from _n′

( )

 onto ′

( )

 . Lemma 2 If 2

( )

f ∈Lα  then the functional

( ) ( )

2 1

( )

, d , ,

f

Tα

ψ

=

∫

_f x

ψ

x x α+ x

ψ

∈ 

is a tempered distribution . Moreover,

( )

2n

f g

Tα =Tα+

 (7) with g

( )

λ =mα+_2n

( )( )

f −λ .

( )

( )

1

ψ

1

ψ

− ₌ − _

 

where ψ λ

( )

=ψ λ

( )

− . So using Theorem 2 we get

( )

( )

( )( )

( )( )

( )( ) ( )

( )( ) ( )

2 1 2 1

1 1

2 4 1 2

2 4 1 ₂

2

, d ( ) d

d

d , ,

f

n n

n _n

n g

T f x x x x f x x x x

m f

m f T

α α

α

α α

α α

α

ψ ψ ψ

λ ψ λ λ λ

λ ψ λ λ λ ψ

+ +

− −

+ + +

+ + +

+

= =

= −

= − =

∫

∫

∫

∫



 





  





which completes the proof.

Lemma 3 Let f ∈_n

( )

 and S∈_n′

( )

 . Then for k=1, 2, we have

( )

k

( ) ( ) ( )( )

k

f

λ

i

λ

f

λ

Λ =

  (8)

( )

k

( ) ( )

k

S iλ S

Λ = −

  (9)

Proof. Identity (8) may be found in [7]. If ψ∈

( )

 then

( )

1

( ) ( )

1

( )

, , 1 k ,

k k k

S ψ S − ψ S − ψ

Λ = Λ = − Λ

  

But by (8),

( )

(

( )

)

1 1 k

k − ψ − _iλ ψ

Λ  =

So

( )

( )

1

(

( )

)

( )

( ) ( )

( ) ( ) ( )

, 1k , k 1k , k 1k k , ,

k

S

ψ

S − i

λ ψ

S i

λ ψ

i

λ

S

ψ

Λ = − = − = −

   

which ends the proof.

Notation 2 From now on assume m=1, 2,. Let _2,m_α be the Sobolev type space constructed by the dif- ferential-difference operator Λ, i.e.,

( )

( )

{

2 2

}

2, : , 1, 2, , .

m _{f L} j_{f L j m}

α = ∈ α  Λ ∈ α  = 



More explicitly, f ∈_2,mα if and only if for each j=1, 2,,m, there is a function in L2α

( )

 abusively denoted by _Λj_{f , such that}

j

j

f _f

Tα T_Λα

Λ = .

Proposition 1 For f∈_2,mα we have

(

Λmf

)

( ) ( )

λ

= i

λ

m

( )( )

f

λ

.

  (10)

Proof. From the definition of _2,m

α

 we have

m

m

f _f

Tα T_Λα

Λ =

By (7) and (9),

(

m

)

( )

m

( )

2n

f f g

Tα i

λ

Tα Tα+

Λ = − =

 

with g

( )

λ =mα+_2n

(

−iλ

)

m

( )( )

f −λ . Again by (7),

( )

2

m

n h f T_Λα =Tα+ 

with h

( )

λ =m_α+2n

(

Λmf

)

( )

−λ . Identity (10) is now immediate.

( ) ( )

(

)

(

)

( )

( )

(

)

(

)

( )

2 2

2 1

1 2 2 2 2

2 2

2 1

1 2 2 2 2

2

1 d

2 _{2 2}

2

1 d

2 _{2 2}

n x

n

n

n

f x y xyt

xy x y

f y A t t

x y xyt

x y xyt

f x y xyt

xy x y

A t t

x y xyt

x y xyt

τ ₋

−

+ −  ₋ 

 

= +

 _{+ −} 

+ − _{ }

− + −  ₋ 

 

+ −

 _{+ −} 

+ − _{ }

∫

∫

where

( )

(

)

(

)

2 1 2

2

2 1 1

n n

A t =a_α+ +t −t α+ −

with a_{α +2n} given by (5).

Proposition 2 Let x∈ and f ∈L2_α

( )

 . Then τxf ∈L2_α

( )

 and 2

2, 2, 2

x n

f _α x f _α

τ ≤ (11) Furthermore,

( )

( )

2

( ) ( )( )

2

x n

n

f x eα i x f

τ λ = ₊ λ λ

  (12)

3. Equivalence of

K-

Functionals and Modulus of Smoothness

Definition 4 Let 2

( )

f ∈L_α  and r>0. Then

 The generalized modulus of smoothness is defined by

(

)

_2,

2, 0

, sup m

m h

h r

f r _α f _α

ω

< ≤

= ∆

where

(

2

)

,

m

m h n

h f τ h I f

∆ = −

I being the unit operator.

 The generalized K-functional is defined by

(

,

)

2, inf

{

2, 2, : 2,

}

.

m m

m

K f r α = f −g α+ Λr g _α g∈α

The next theorem, which is the main result of this paper, establishes the equivalence between the generalized modulus of smoothness and the generalized K-functional:

Theorem 4 There are two positive constants c₁=c m n₁

(

, ,α

)

and c₂=c₂

(

m n, ,α

)

such that

(

)

2

(

)

(

)

1 , 2, , _2, 2 , 2,

mn m

m m m

c f r _α r K f r c f r _α

α

ω ≤ ≤ ω

for all f ∈L_α2

( )

 and r>0.

In order to prove Theorem 4, we shall need some preliminary results.

Lemma 4 Let f ∈Lα2

( )

 and h>0. Then

2 2, 2, 3

m m mn

hf _α h f α

∆ ≤ (13)

(

)

( )

2

(

( )

)

( )( )

2 1

m

m mn

h f λ h eα+ n i hλ f λ

∆ = −

  (14)

Proof. The result follows easily by using (11), (12) and an induction on m.

Lemma 5 For all f ∈_2,mα and h>0 we have

(2 1)

2, 2,

m n

m m

hf _α h f _α

+

Proof. By (10), (14), Lemma 1 (4) and Theorem 2 we have

(

)

( )

( )

( )

( )( )

( )

( )

_{( )( )}

_{( )}

( )

(

)

_{( )}

_{( )}

( )

2 2

2,

2 2

4

2

2 2

2 2 1

2 2 2 1

2 2 2 1

2, d

1 d

d

d

m m

h h

m mn

n

m m n

m n m

m n m

f f

h e i h f

h f

h f

h f

α

α

α

λ σ λ

λ λ σ λ

λ λ σ λ

λ σ λ

+ + + +

∆ = ∆

= −

≤

= Λ

= Λ

∫

∫

∫

∫







 







which is the desired result.

Notation 3 For 2

( )

f ∈L_α  and ν >0 define the function

( )( )

2

( )( )

( ) ( )

2 d

n

n

Pν f x =x

∫

₋ν_ν f λ eα+ i xλ σ λ Proposition 3 Let 2

( )

f∈L_α  and ν >0. Then

1) The function Pν

( )

f is infinitely differentiable on  and

( )( )

2

( )( )( )

( ) ( )

2 d

k

k n

n

Pν f x x ₋ν_ν f λ λi eα+ i xλ σ λ

Λ =

_∫

 (16)

for all k=0,1,.

2) For all k=0,1,, k

( )

2

P_ν f L_α

Λ ∈ and

( )

(

k

)

( ) ( ) ( )( ) ( )

k ,

P_ν f

λ

i

λ

f

λ χ λ

_ν

Λ =

  (17)

where

( )

1 if ,

0 if .

ν

λ ν χ λ

λ ν

 ≤

 = 

> 

Proof. The fact that P_ν

( )

f ∈∞

( )

 follows from the derivation theorem under the integral sign. Identity (16) follows readily from (2) and the relationship

(

2

)

2

2

n n

n x f x Dα+ f

Λ =

which is proved in [7]. Assertion (2) is a consequence of (16) and Theorem 2.

Lemma 6 There is a positive constant c=c

(

α,n

)

such that

( )

2 1

2, 2,

m mn m

f −Pν f _α ≤c− ν ∆ νf _α

for any f ∈L2α

( )

 and ν >0.

Proof. By (17) and Theorem 2, we have

( )

2

( )

2

( )( )

2

( )

( )( )

2

( )

2, 1 d d

f P_ν f _{α ν} f f

λ ν

χ λ λ σ λ _≥ λ σ λ

− =

_∫

− =

_∫

  

By Lemma 1 (2) there is a constant c>0 which depends only on

α

and n such that

(

)

2

1−e_{α+ n} iλ ν ≥c

for all λ ∈ with λ ν≥ . From this, (14) and Theorem 2 we get

( )

(

)

( )( )

( )

(

)

( )

( )

2 ₂ 2 2

2 2,

2 2

2 4 2 4

1 1

2,

1 d

d

m m

n

m mn m m mn m

f P f c e i f

c f c f

ν _{α λ ν} α

ν ν _α

λ ν λ σ λ

ν λ σ λ ν

−

+ ≥

− −

− ≤ −

= ∆ = ∆

∫

∫





which achieves the proof.

Corollary 1 For all f ∈L2α

( )

 and ν >0 we have

( )

2

(

)

2,

2, ,1

m mn m

f −P_ν f _α ≤c−

ν ω

f

ν

_α

where c is as in Lemma 6.

Lemma 7 There is a positive constant C=C

(

α,n

)

such that

( )

(2 1)

1

2, 2,

m n

m_{P f C}m m _f

ν _α ν + ν _α

Λ ≤ ∆

for every 2

( )

f∈L_α  and ν >0.

Proof. By (17) and Theorem 2 we have

( )

( )( )

( )

(

)

(

)

( )( )

( )

2 ₂ 2

2, 2 2 2 2 2 2 d 1 d 1 m m m m n m n

P f f

e i f

e i

ν

ν _{α ν}

ν

α ν

α

λ λ σ λ

λ _{λ ν λ σ λ}

λ ν − + − + Λ = = − −

∫

∫

  Put

( )

1 ₂ . sup 1 t _n t C

e_α it

≤ ₊

= −

By L’Hôpital’s rule,

( )

(

)

0 2

lim 2 2 1 .

1 t

n

t

n

e_α it

α

→ +

= + +

−

This when combined with Lemma 1 (3) entails 0<C<∞. Moreover,

(

)

(

)

(

)

( )

( )

2 2 2 2 2 2 2 2 2 2 2 1 2 sup sup 1 1 sup 1 m m m m m n n m m m m t n

e i e i

t

C

e it

λ ν _α λ ν _α

α

λ ν

λ _ν

λ ν λ ν

ν ν ≤ ≤ + + ≤ ₊ = − − = = − Therefore

( )

( )

(

)

( )( )

( )

( )

(

)

_{( )}

_{( )}

( )

(

)

_{( )}

_{( )}

( )

2 2 2 2

2 2,

2 2 2 1

2

1

2 2 2 1

2

1

2 2 2 1

2 1 _2, 1 d d d m m m n m n m m m n m m m n m m

P f C e i f

C f

C f

C f

ν

ν _{α ν} α

ν ν ν

ν

ν _α

ν λ ν λ σ λ

ν λ σ λ

ν λ σ λ

ν + − + − + +

Λ ≤ −

= ∆ ≤ ∆ = ∆

∫

∫

∫

   

by virtue of (14) and Theorem 2.

Corollary 2 For any 2

( )

f ∈L_α  and ν >0 we have

( )

(2 1)

₍

₎

2,

2, ,1

m n

m m

m

Pν f _α C ν ω f ν α

+

Λ ≤

where C is as in Lemma 7.

(

)

( ) ( )

(

)

2, 2, 2,

2 1 2

2, _2,

2 1 2

2, _2,

2

2, _2,

3 3 3

m m m

h h h

m n

m mn m

m n

m mn m

m mn m m

f f g g

h f g h g

r f g r g

r f g r g

α α α

α α

α α

α α

+

+

∆ ≤ ∆ − + ∆

≤ − + Λ

≤ − + Λ

≤ − + Λ

Calculating the supremum with respect to h∈

] ]

0,r and the infimum with respect to all possible functions

2, m

g∈_α we obtain

(

)

2

(

)

1 , 2, , 2,

mn m

m m

c f r _α r K f r

α

ω ≤

with _c1=3−m_.

2) Let

ν

be a positive real number. As Pν

( )

f ∈_2,mα it follows from the definition of the K-functional and Corollaries 1 and 2 that

(

)

( )

( )

(

)

(

)

(

)

( )

(

)

(

)

2, 2,

2,

2 (2 1)

2, 2,

2

2, 2,

,

,1 ,1

,1 ,1 .

m m m

m

m mn m m m n

m m

m

mn m m

m m

K f r f P f r P f

c f C r f

c f C r f

ν _α ν _α

α

α α

α α

ν ω ν ν ω ν

ν ω ν ν ω ν

− +

−

≤ − + Λ

≤ +

≤ +

Since

ν

is arbitrary, by choosing ν =1r we get

(

)

(

)

2

2 2,

2,

, ,

mn m

m m

r K f r c f r _α

α ≤ ω with _c2 =_c−m+_Cm_{. This concludes the proof.}

References

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[

0,+∞

[

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