Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.
If we wish to apply the Laplacian to a polynomial, we apply it to each term of a homogeneous polynomial (noting that the Laplacian is linear).
A homogeneous harmonic polynomial is a spherical harmonic.
A spherical harmonic is a restriction to the unit sphere of homogeneous harmonic polynomials of degree $n.$
A function on the sphere is harmonic.
You can recover it if you know the spherical harmonics.
Fourier series on the $n-$dimensional sphere are in terms of spherical harmonics.
