Series Analysis and Schwartz Algebras of Spherical Convolutions on Semisimple Lie Groups

We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(\lambda ),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^p-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^p-$ Schwartz functions.