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.
Boundedness of projection operators and Cesàro means in weighted $L^p$ space on the unit sphere
On spherical expansions of smooth $\gr{SU}{n}$-zonal functions on the unit sphere in $\NC^n$
Maximal function and Multiplier Theorem for Weighted Space on the Unit Sphere
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