Mean convergence of Fourier-Dunkl series

TL;DR

This paper investigates the convergence properties of Fourier-Dunkl series in weighted norms, establishing conditions on weights for convergence and identifying necessary and sufficient criteria.

Contribution

It provides new conditions based on Muckenhoupt's $A_p$ classes that guarantee the weighted norm convergence of Fourier-Dunkl series, including necessary and sufficient criteria.

Findings

Weighted norm convergence is characterized by $A_p$ weight conditions.

Necessary and sufficient conditions for convergence are established.

Results extend classical Fourier analysis to Dunkl transform context.

Abstract

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the $A_p$ classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.