Degree-indexed spherical harmonic spaces in three dimensions

Let ${\mathbb{S}}^2=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }=1\}$ with its usual surface-area measure $dS$, normalized so ${\int }_{{\mathbb{S}}^2}dS=4\pi$. For an integer $\mathrm{\ell }\ge 0$, let ${\mathcal{H}}_{\mathrm{\ell }}$ be the complex vector space of restrictions to ${\mathbb{S}}^2$ of homogeneous degree-$\mathrm{\ell }$ polynomials $P(x_1,x_2,x_3)$ satisfying $\mathrm{\Delta }P=0$. Give ${\mathcal{H}}_{\mathrm{\ell }}$ the inner product $⟨f,g⟩={\int }_{{\mathbb{S}}^2}f\bar{g}dS$, linear in the first argument. An orthonormal basis is not fixed by this definition; the addition-theorem item proves that $\dim {\mathcal{H}}_{\mathrm{\ell }}=2\mathrm{\ell }+1$ and that such a finite basis exists.

If $Y_{\mathrm{\ell }m}\in {\mathcal{H}}_{\mathrm{\ell }}$ is later chosen from an orthonormal basis and $v\mathrm{\ne }0$, its associated homogeneous harmonic polynomial is $H_{\mathrm{\ell }m}(v)=\mathrm{\mid }v{\mathrm{\mid }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell }m}(v\mathrm{/}\mathrm{\mid }v\mathrm{\mid })$. At $v=0$ use the polynomial value: $H_{00}(0)=Y_{00}$ and $H_{\mathrm{\ell }m}(0)=0$ for $\mathrm{\ell }>0$. This is the regular-solid-harmonic convention used in the 3D evaluator.