Let $f\in L^{2}(S^{n-1})$ where $S^{n-1}$ is the unit sphere of $\Bbb{R}^n$.
As it is known $f$ has the following spherical Harmonics expansion (the convergence is $L^{2}(S^{n-1})$ in :
$f(w)=\sum_{j} Y_j(w)$ where $Y_j\in H_j$ with $H_j$ is the space of spherical harmonics of degree $j$.
Now let $f\in C(S^{n-1})$ (space of continuous functions on S^{n-1} ) then $f\in L^{2}(S^{n-1})$.
Hence $f(w)=\sum_{j} Y_j(w)$.
My question can we say that the series in question converge uniformly and absolutely on $S^{n-1}$.