Harmonic decomposition of homogeneous polynomials (${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$)

If ${\mathcal{P}}_{\ell }$ is the space of degree-$\ell$ homogeneous polynomials and ${\mathcal{H}}_{\ell }$ its harmonic polynomials, then ${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$. Under the Fischer inner product, the second summand is orthogonal to the kernel of the Laplacian. Iteration expresses every polynomial restricted to the unit sphere as a sum of spherical harmonics.