New Results on the Equivalence of -Functionals and Modulus of Continuity of Functions Defined on the Sobolev Space Constructed by the Generalized Jacobi-Dunkl Operator

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.