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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.

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

where

which is referred to as the Dunkl operator with parameter

The authors have developed in [

In the classical theory of approximation of functions on

In addition to modulus of smoothness, the K-functionals introduced by J. Peetre [

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

The one-dimensional Dunkl kernel is defined by

where

is the normalized spherical Bessel function of index

Furthermore, we have the Laplace type integral representations:

where

The following properties will be useful for the sequel.

Lemma 1 1) For all

2) There is

3) For all

4) For all

Proof. Assertions (1) and (2) are proved in [

we have

Clearly the integral above is null only for

we get

By (6),

Moreover,

which concludes the proof.

Notation 1 Put

We denote by

The topology of

Clearly

For

Definition 1 The generalized Fourier transform of a function

Remark 1 If

Theorem 1 The generalized Fourier transform

where

Theorem 2 1) For every

2) The generalized Fourier transform

Definition 2 The generalized Fourier transform of a distribution

Theorem 3 The generalized Fourier transform

Lemma 2 If

is a tempered distribution

with

Proof. The fact that

where

which completes the proof.

Lemma 3 Let

Proof. Identity (8) may be found in [

But by (8),

So

which ends the proof.

Notation 2 From now on assume

More explicitly,

Proposition 1 For

Proof. From the definition of

By (7) and (9),

with

with

Definition 3 The generalized translation operators

where

with

Proposition 2 Let

Furthermore,

Definition 4 Let

The generalized modulus of smoothness is defined by

where

I being the unit operator.

The generalized K-functional is defined by

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

for all

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

which is the desired result.

Notation 3 For

Proposition 3 Let

1) The function

for all

2) For all

where

Proof. The fact that

which is proved in [

Lemma 6 There is a positive constant

for any

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

By Lemma 1 (2) there is a constant

for all

which achieves the proof.

Corollary 1 For all

where c is as in Lemma 6.

Lemma 7 There is a positive constant

for every

Proof. By (17) and Theorem 2 we have

Put

By L’Hôpital’s rule,

This when combined with Lemma 1 (3) entails

Therefore

by virtue of (14) and Theorem 2.

Corollary 2 For any

where C is as in Lemma 7.

Proof of Theorem 4. 1) Let

Calculating the supremum with respect to

with

2) Let

Since

with