I don't know if I can ask this question here, but there's a question on MO for which I have a good interest. The problem is I don't think I have competencies to do it. On the page The spherical harmonics are the EIGENVECTORS of Beltrami operator, Paul Siegel has answered this question. Could anyone be able to develop in details his answer?
Interest question : Is every eigenfunction of the Laplacian a spherical harmonic?
Questions : ($1$) Why if $H$ is dense in $L^2 (\mathbb{S}^2)$, then that answer respond the main question?
($2$) Why every polynomial of $\mathbb{S}^2$ is a sum of homogeneous polynomial?
($3$) Why every homogeneous polynomial is a sum of harmonic polynomial?
($4$) Why a spherical harmonic is in fact an harmonic polynomial restricted to $\mathbb{S}^2$? I have tried with the spherical harmonic $u(x,y,z)=xyz$ and use this spherical coordinates, but it seems don't respect the homogeneous property.
Thanks!