Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ is the dimension of the subspace of spherical harmonics of degree $j$.
Consider expansion of $f$ on spherical harmonics: $$f=\sum_{j,k}f_{j,k}Y_{j,k},$$ where $f_{j,k}$ are Fourier-Laplace coefficients of $f$.
What would be the expansion of $f^2$ on spherical harmonics $Y_{j,k}$?