#### Abstract

In this paper, we establish some new generalized results on the equivalence of -functionals and modulus of continuity of functions defined on the Sobolev space , by using the harmonic analysis related to the Jacobi-Dunkl operator , where and .

#### 1. Introduction

The modulus of continuity plays a fundamental role in the approximation theory of functions. For a given positive real number and a positive integer , the classical modulus of continuity of a function is defined by where , such that is the unit operator in and represents the usual translation operator given by .

The classical -functional, introduced by Peetre in [1], is defined by where is the Sobolev space constructed by the operator and

A remarkable result of the approximation theory of functions on which establishes the equivalence between the modulus of continuity and the -functional can be formulated as follows:

Theorem 1 (see [2]). *There exist two positive real constants and such that for any function and , we have
*

The translation operator is used for the construction of modulus of continuity and smoothness which are the fundamental elements in the classical theory of the approximation of functions on (see [3–5]).

The equivalence between the modulus of continuity and the -functionals has been established in [6]. Various generalized continuity moduli are studied in [7, 8]. A considerable attention has been devoted to finding generalizations of Theorem 1 (see, for example, [9–12]). This theorem was recently generalized in [13], for the Jacobi transform in the space by using a generalized Jacobi translation operator. Many generalized moduli 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, for example, [14]).

Motivated by the results mentioned above, in this paper, we establish an analogue result of Theorem 1 by using the harmonic analysis associated with the Jacobi-Dunkl operator. The usual translation operators are replaced by generalized Jacobi-Dunkl translation operators in our new result.

Our paper is organized as follows: Section 2 gives some basic definitions, notations, lemmas, and theorems as preliminaries. Our main result is given in Section 3 followed by conclusion and future works in Section 4.

#### 2. Preliminaries

In this section, we give some notations, definitions, and results of the harmonic analysis related to the Jacobi-Dunkl operator defined by where and . We cite here, as briefly as possible, only those properties actually required for the discussion; for more details, we refer the reader to [15, 16].

The Jacobi-Dunkl Laplacian operator defined on is given by which is equivalent to where is the Jacobi operator defined by

In the rest of the paper, we denote the following: (i): the space of indefinitely differentiable functions on with compact support(ii): the space of indefinitely differentiable pair functions on with compact support(iii): the usual Schwartz space of indefinitely differentiable functions rapidly decreasing on , as well as their derivatives of all orders, provided with the topology of the following seminorms

This space is provided with the topology of the following seminorms , , where (iv), , and the space of measurable functions on such thatwhere (v) is the Sobolev space constructed by the generalized Laplacian Jacobi-Dunkl operator, i.e.,where ; , .

Let and . (vi)The generalized modulus of continuity of the function at order is defined by the following formula:where and is the unit operator of the space (vii)Gauss’s hypergeometric function is defined by

where , , and is the Pochhammer symbol given by where is the set of nonzero positive integers. (viii)The Jacobi function is defined by(ix)The generalized -functional of a function is defined by

where is the generalized Jacobi-Dunkl Laplacian operator defined by (6).

For the short form, we denote (x)The Jacobi-Dunkl kernel is the unique solution of the following differential equation:

which is defined by

where , , and is the Jacobi function given by (18). (xi)The Jacobi-Dunkl transform of a function is defined by

*Property 2 (see [17]). *(1)If , then and we have(2)If , then

Theorem 3 (see [17]). (i)*The Jacobi-Dunkl transform is a topological isomorphism between and *(ii)*The Jacobi-Dunkl transform is a topological isomorphism from to , and we have*

Theorem 4 (see [17]). *Let such that ; then, we have
*

*Definition 5. *Let and ; we define the function by

where is the characteristic function defined by

and is the inverse Jacobi-Dunkl transform.

*Remark 6. *We can easily prove that the function is indefinitely differentiable and , .

Lemma 7 (see [18]). *Let and ; then,
*(1)* for all *(2)

Lemma 8 (see [18]). *Let and ; then,
*

#### 3. Main Result

In this section, we give the main result of our paper which is a generalization of Theorem 1.

Theorem 9. *There exist two positive real constants and such that for any function and positive constant , we have
**In order to prove Theorem 9, we will need some preliminary results.*

Proposition 10. *If and , then
*

*Proof. *By using (18) and (23), we obtain
We use the proof by induction for , and we obtain the result.

Proposition 11. *Let and ; then, we have
*

*Proof. *Suppose that . By using Lemmas 7 and 8, we have
thus,
By taking the supremum on , we obtain the result.☐

Proposition 12. *Let and ; then, there exists a positive number such that
*

*Proof. *We use the Plancherel formula (26); we have
We use ; then, we have
There exists such that
for such that , and by using Lemma 8, we have

Proposition 13. *Let and .**There exists a positive number such that
*

*Proof. *We use the Plancherel formula (26) and Lemma 7; we have
For , we obtain
Note that if is large enough, then
thus
Finally, formula (42) is proven.

*Remark 14. *Let and . There exists a positive constant such that

*Proof of Theorem 9. *(i)We start by giving the proof of the inequality:For this purpose, let and .

By using Propositions 10 and 11, we have
We calculate the supremum on and the infimum on ; we deduce that
where .
(ii)Now, we give the proof of the inequality:Since , then by using the definition of -functional, we obtain
We use Remark 14; we have
Since is a positive arbitrary value, then by setting , we obtain
where .

#### 4. Conclusion

In this paper, we generalized and we proved some new results on the equivalence between the -functionals and the modulus of continuity of functions defined on the Sobolev space by using the harmonic analysis related to the Jacobi-Dunkl operator.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.