The main theme of this paper is the approximation on the sphere by weighted sums of spherical harmonics. We give necessary and sufficient conditions on the weights for convergence in both the continuous and the cases. Approximation by spherical convolution is a particular and important case that fits into our setting.
V. A. Menegatto. A. C. Piantella. "Approximation on the sphere by weighted Fourier expansions." J. Appl. Math. 2005 (4) 321 - 339, 11 December 2005. https://doi.org/10.1155/JAM.2005.321