# Approximation theorems connected with differential-difference operator

## TL;DR

This paper extends Dunkl theory by developing a generalized Taylor formula with estimates for the integral remainder, and characterizes Besov-type spaces related to the Dunkl operator on the real line.

## Contribution

It introduces an extension of translation and Taylor's formula within Dunkl theory, providing new estimates and space characterizations.

## Key findings

- Derived properties and estimates of the integral remainder in the generalized Taylor formula.
- Described Besov-type spaces with specified remainder order.
- Extended Dunkl theory concepts to include translation and approximation tools.

## Abstract

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.