Intertwining operator associated to symmetric groups and summability on the unit sphere

TL;DR

This paper derives an integral representation for the intertwining operator related to Dunkl operators for symmetric groups, leading to explicit formulas for reproducing kernels and results on Cesàro summability of h-harmonic series on the sphere.

Contribution

It provides a new integral formula for the intertwining operator and explicit reproducing kernels, enabling sharp Cesàro summability results for h-harmonics on the sphere.

Findings

Explicit integral representation of the intertwining operator.

Closed-form formulas for reproducing kernels of h-harmonics.

Sharp Cesàro summability results for h-harmonic series.

Abstract

An integral representation of the intertwining operator for the Dunkl operators associated with symmetric groups is derived for the class of functions of a single component. The expression provides a closed form formula for the reproducing kernels of $h$-harmonics associated with symmetric groups when one of the components is a coordinate vector. The latter allows us to establish a sharp result for the Ces\`aro summability of $h$-harmonic series on the unit sphere.