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Approximation theorems connected with differential-difference operator

In the present paper, we propose to give an extension to the context of Dunkl theory of the notion of translation and in connection with this a corresponding extension of Taylor's formula. More precisely, we prove some properties and estimates of the integral remainder in the generalized Taylor formula associated to the Dunkl operator on the real line and we describe the Besov-type spaces for which the remainder has a given order.


Introduction
Delsartes gave in [10,11] a certain extension of the notion of translation and in connection with this a corresponding extension of Taylor's formula. Generalized translation have later been considered from various points of view by many authors (see Levitan [13], Bochner [8,9]). Löfström and Peetre in [14] estimated the remainder in the generalized Taylor's formula and they described the space of functions for which the remainder has a given order.
Our aim in this paper is to extend the results obtained in [14] to the context of Dunkl theory. More precisely, we prove some properties and estimates of the integral remainder of order k associated to the Dunkl operator on the real line and we establish the coincidence between two characterizations of Besov-type spaces related to this integral remainder.
For a real parameter α > − 1 2 , the Dunkl operator on the real line denoted by Λ α , is a differential-difference operator introduced in 1989 by C. Dunkl in [12]. This operator is associated with the reflection group Z 2 on R and is given by The Dunkl operator can be considered as a perturbation of the usual derivative by reflection part. This operator plays a major role in the study of quantum harmonic oscillators governed by Wigner's commutation rules (see [18]). The Dunkl kernel E α related to Λ α is used to define the Dunkl transform which enjoys properties similar to those of the classical Fourier transform. The Dunkl kernel E α satisfies a product formula (see [19]). This allows us to define the Dunkl translation τ x , x ∈ R (see next section). If the parameter α = − 1 2 , then the operator Λ α reduces to the differential operator df dx . Therefore Dunkl analysis can be viewed as a generalization of the classical Fourier analysis on R.
In 2003, the classical Taylor formula with integral remainder was extended in [16] to the one dimensional Dunkl operator Λ α : For k = 1, 2, ..., f ∈ E(R) and a ∈ R, we have is the integral remainder of order k given by where E(R) is the space of infinitely differentiable functions on R and (Θ p ) p∈N , (b p ) p∈N are two sequences of functions constructed inductively from the function A α defined on R by A α (x) = |x| 2α+1 (see next section).
There are many ways to define the Besov spaces (see [6,7,17]) and the Besov-Dunkl spaces (see [1,2,3,4]). It is well known that Besov spaces can be described by means of differences using the modulus of continuity of functions. These spaces defined by the modulus of smoothness occur more naturally in many areas of analysis including approximation theory.
• We denote by L p (µ α ) the space of complex-valued functions f , measurable on R such that where µ α is a weighted Lebesgue measure associated to the Dunkl operator given by with A α is the function defined on R by where K k p,α is the Peetre K-functional given by The contents of the present paper are as follows. In section 2, we collect some basic definitions and results about harmonic analysis associated with the Dunkl operator Λ α . In section 3, we give some properties and estimates of the integral remainder of order k. Finally, we establish that Along this paper, we use c to represent a suitable positive constant which is not necessarily the same in each occurrence.

Preliminaries
In this section, we recall some notations and results in Dunkl theory on R and we refer for more details to [5,12,19].
For λ ∈ C, the initial problem has a unique solution E α (λ.) called Dunkl kernel given by where j α is the normalized Bessel function of the first kind and order α. The Dunkl kernel E α satisfies the following product formula where γ x,y is a signed measure on R with compact support. For x, y ∈ R and f a continuous function on R, the Dunkl translation operator τ x is given by . The Dunkl translation operator satisfies the following properties : (1) τ x is a continuous linear operator from E(R) into itself.
(2) For all f ∈ E(R), we have (3) For all x ∈ R, the operator τ x extends to L p (µ α ), p ≥ 1 and we have for f in L p (µ α ) It has been shown in [16], the following generalized Taylor formula with integral remainder: For k = 1, 2, ..., f ∈ E(R) and a ∈ R, we have is the integral remainder of order k given by According to ([20], Lemma 2.2), the Dunkl operator Λ α have the following regularity properties: Λ α leaves C ∞ c (R) and the Schwartz space S(R) invariant. (2.5)

Characterizations of Besov-Dunkl spaces of order k
In this section, we start with the proof of some properties and estimates of the integral remainder in the generalized Taylor formula. (1) From Proposition 2.1, we have

2)
(3) Note that the function y −→ τ y (f ) − f is continuous on R (see [15], Lemma 1, (ii)), which implies that the same is true for the function y −→ R k (y, f ).
Before establishing that B k D β,α p,q = K k D β,α p,q , we give a remark, a proposition containing sufficient conditions and an example.
Example 3.1. From (2.5) and Proposition 3.1, we can assert that the spaces C ∞ c (R) and S(R) are included in B k D β,α p,q .