Recall that, for fixed $k \in Z_+\cup \{0\}$, a spherical harmonic of degree $k$ is the restriction to $S^{n-1}$ of a harmonic polynomial on $R^n$ that is homogeneous of degree $k$. Let $Y_{k,1}\, ,Y_{k,2}\,\ldots, Y_{k,d_{n-1}(k)}$, with $\displaystyle{d_{n-1}(k):={n+k-1\choose k}}$, be an orthonormal basis of the subspace in $L^{2}(S^{n-1})$ of degree $k$ of spherical harmonics. Then, $\{Y_{k,m}\}$ is an orthonormal basis for $L^2(S^{n-1})$, i.e. for any $g\in L^2(S^{n-1})$, one has: $$ g=\sum_{k,m}g_{k,m}Y_{k,m}, $$ with $$ g_{k,m}:=\int_{S^{n-1}}g(\omega)Y_{k,m}(\omega) d\omega, \quad k=0,1, \ldots; m=1,2, \ldots, d_{n-1}(k), $$ This so-called spherical harmonic expansion of $g$ converging to it in $L^2(S^{n-1})$.
QUESTION: How to find high order derivative $g^{(\ell)}$.