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

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


Introduction
Consider the first-order singular differential-difference operator on the real line where 1 2 α > − and 0,1, n =  .For 0 n = , we regain the differential-difference operator ( ) ( ) ( ) ( ) which is referred to as the Dunkl operator with parameter 1 2 α + associated with the reflection group 2  on  .Such operators have been introduced by Dunkl [1]- [3] in connection with a generalization of the classical theory of spherical harmonics.The one-dimensional Dunkl operator D α 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-difference 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 ( ) . 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 ( ) As the notion of translation operators was extended to various contexts (see [9] [10] and the references therein), many generalized modulus of smoothness have been discovered.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 0 n = .

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 1 2

Equivalence of K-Functionals and Modulus of Smoothness
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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 , , , , In order to prove Theorem 4, we shall need some preliminary results.Lemma 4 Let ( ) Proof.The result follows easily by using (11), (12) and an induction on m.Lemma 5 For all Proof.By ( 10), ( 14), Lemma 1 (4) and Theorem 2 we have ( )( 1) The function ( ) where ( ) Proof.The fact that ( ) ( ) follows from the derivation theorem under the integral sign.Identity (16) follows readily from (2) and the relationship ( ) which is proved in [7].Assertion (2) is a consequence of ( 16) and Theorem 2. Lemma 6 There is a positive constant ( ) Proof.By (17) and Theorem 2, we have By Lemma 1 (2) there is a constant 0 c > which depends only on α and n such that where c is as in Lemma 6. Lemma 7 There is a positive constant ( ) Proof.By (17) and Theorem 2 we have . sup 1   ∈  it follows from the definition of the K-functional and Corollaries 1 and 2 that ( )

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

I
being the unit operator. The generalized K-functional is defined by all λ ∈  with λ ν ≥ .From this, (14) and Theorem 2 we get the proof. )