About the almost everywhere convergence of the spectral expansions of   functions from $L_1^\a(S^N)$
  In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space $L_1^\a$ is used.   For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we proved the positive results on summability of Fourier-Laplace series. Note that when order $\alpha$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series, which based on spectral expansions property of self-adjoint Laplace-Beltrami operator on the unit sphere. 
Anvarjon Akhmedov