# Fractional smoothness in $L^p$ with Dunkl weight and its applications

## TL;DR

This paper introduces fractional smoothness concepts in Dunkl-weighted $L^p$ spaces, establishing approximation theorems and inequalities for entire functions of spherical exponential type.

## Contribution

It defines fractional Dunkl Laplacian, modulus of smoothness, and $K$-functional in weighted spaces, advancing approximation theory in this fractional Dunkl setting.

## Key findings

- Proved direct and inverse approximation theorems.
- Established inequalities for entire functions of spherical exponential type.
- Extended classical approximation results to fractional Dunkl contexts.

## Abstract

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some inequalities for entire functions of spherical exponential type in fractional settings.