Spherical harmonics form a complete basis with the addition formula

If $P_{\mathrm{\ell }}(x)=r^{\mathrm{\ell }}Y(\omega )$ is homogeneous and harmonic, the polar Laplacian convention in [F1] gives $0=\mathrm{\Delta }{P}_{\mathrm{\ell }}=r^{\mathrm{\ell }-2}(\mathrm{\ell }(\mathrm{\ell }+1)Y+{\mathrm{\Delta }}_{S^2}Y)$. Hence $-{\mathrm{\Delta }}_{S^2}Y=\mathrm{\ell }(\mathrm{\ell }+1)Y$.

Let ${\mathcal{P}}_n$ be the homogeneous complex polynomials of degree $n$ in three variables, with monomial basis of size $\left(\genfrac{}{}{0px}{}{n+2}{2}\right)$. For homogeneous $R_m$ of degree $m$, direct differentiation gives $\mathrm{\Delta }(\mathrm{\mid }x{\mathrm{\mid }}^{2a}{R}_m)=2a(2m+2a+1)\mathrm{\mid }x{\mathrm{\mid }}^{2a-2}{R}_m+\mathrm{\mid }x{\mathrm{\mid }}^{2a}\mathrm{\Delta }{R}_m$. Given $S\in {\mathcal{P}}_{n-2}$, put $J=\lfloor (n-2)\mathrm{/}2\rfloor$, $c_j=2(j+1)(2n-2j-1)$, $b_0=1\mathrm{/}{c}_0$, and $b_j=-b_{j-1}\mathrm{/}{c}_j$ for $1\le j\le J$. Then $Q={\sum }_{j=0}^J{b}_j\mathrm{\mid }x{\mathrm{\mid }}^{2(j+1)}{\mathrm{\Delta }}^{j}S\in {\mathcal{P}}_n$ satisfies $\mathrm{\Delta }Q=S$: the coefficient of $\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j}S$ is $b_j{c}_j+b_{j-1}=0$ for $j\ge 1$, the $S$ coefficient is $b_0{c}_0=1$, and ${\mathrm{\Delta }}^{J+1}S=0$. Thus $\mathrm{\Delta }:{\mathcal{P}}_n\to {\mathcal{P}}_{n-2}$ is onto. By [F5], its harmonic kernel has dimension $\left(\genfrac{}{}{0px}{}{n+2}{2}\right)-\left(\genfrac{}{}{0px}{}{n}{2}\right)=2n+1$ for $n\ge 2$; for $n=0,1$ all homogeneous polynomials are harmonic and have dimensions $1,3$. Restriction to $S^2$ is injective because homogeneity recovers $P(r\omega )=r^{n}P(\omega )$, so $\dim {\mathcal{H}}_n=2n+1$.

Let $P_{\mathrm{\ell }},Q_m$ be homogeneous harmonic polynomials of degrees $\mathrm{\ell },m$. Apply [F2] on the unit ball to the real and imaginary components of $P_{\mathrm{\ell }},\overline{Q_m}$. Their volume Laplacians vanish, while homogeneity gives outward normal derivatives ${\mathrm{\partial }}_{\nu }{P}_{\mathrm{\ell }}=\mathrm{\ell }{P}_{\mathrm{\ell }}$ and ${\mathrm{\partial }}_{\nu }\overline{Q_m}=m\overline{Q_m}$ on $S^2$. Therefore $(\mathrm{\ell }-m){\int }_{S^2}{P}_{\mathrm{\ell }}(\omega )\overline{Q_m(\omega )}d\sigma (\omega )=0$. If $\mathrm{\ell }\mathrm{\ne }m$, this proves orthogonality.

For $P\in {\mathcal{P}}_n$, define $H=P+{\sum }_{j=1}^{\lfloor n\mathrm{/}2\rfloor }{a}_j\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j}P$, where $a_0=1$ and $a_{j+1}=-a_j\mathrm{/}[2(j+1)(2n-2j-1)]$. The differentiation identity in step 1.2 shows that the coefficients of each $\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j+1}P$ cancel in $\mathrm{\Delta }H$, so $H$ is harmonic. Since $P-H$ is divisible by $\mathrm{\mid }x{\mathrm{\mid }}^2$, restriction to $S^2$ writes $P{\mathrm{\mid }}_{S^2}$ as an element of ${\mathcal{H}}_n$ plus the restriction of a homogeneous degree-$(n-2)$ polynomial. Induction on degree decomposes every polynomial restriction into a finite sum of spherical harmonics.

Fix $\mathrm{\ell }$. By [F10], the $L^2$ inner product is positive definite on the finite-dimensional space of continuous restrictions, so finite-dimensional Gram--Schmidt gives an orthonormal basis. The sum $Z_{\mathrm{\ell }}$ is independent of this basis because a unitary change of basis preserves the sum of rank-one coordinate kernels. For $R\in SO(3)$ and Borel $E\subseteq S^2$, the polar-surface definition represents $\sigma (E)$ as three times the Lebesgue measure of its unit radial cone $C_E=\{r\omega :0<r\le 1,\omega \in E\}$. Since $R$ is a homeomorphism, $RE$ is Borel, $R{C}_E=C_{RE}$, and $\mathrm{\mid }\det R\mathrm{\mid }=1$; [F7] therefore gives $\sigma (RE)=\sigma (E)$. If $P$ is homogeneous and harmonic of degree $\mathrm{\ell }$, then $P\circ R^{-1}$ is homogeneous of the same degree and remains harmonic: termwise differentiation of monomials in the real and imaginary parts gives ${\mathrm{\partial }}_i{\mathrm{\partial }}_i(P\circ R^{-1})(x)={\sum }_{a,b}(R^{-1}{)}_{ai}(R^{-1}{)}_{bi}({\mathrm{\partial }}_a{\mathrm{\partial }}_{b}P)(R^{-1}x)$; summing over $i$ and using $R^{-1}(R^{-1}{)}^T=I$ proves $\mathrm{\Delta }(P\circ R^{-1})=(\mathrm{\Delta }P)\circ R^{-1}=0$. Thus composition with $R^{-1}$ is unitary on $L^2(S^2,d\sigma )$ and preserves ${\mathcal{H}}_{\mathrm{\ell }}$; the rotated basis is again orthonormal, so basis independence gives $Z_{\mathrm{\ell }}(R\omega ,R\eta )=Z_{\mathrm{\ell }}(\omega ,\eta )$. In particular $Z_{\mathrm{\ell }}(\omega ,\omega )$ is constant because rotations act transitively on $S^2$. Integrating the diagonal gives ${\int }_{S^2}{Z}_{\mathrm{\ell }}(\omega ,\omega )d\sigma =2\mathrm{\ell }+1$. Polar coordinates applied to the unit ball and its volume $4\pi \mathrm{/}3$ (A closed three-dimensional ball of radius $r\ge 0$ has volume $4\pi r^3\mathrm{/}3$) give $\sigma (S^2)=4\pi$, so the diagonal constant is $(2\mathrm{\ell }+1)\mathrm{/}(4\pi )$. Cauchy–Schwarz for the finite coefficient vector yields the displayed evaluation bound.

Let $A$ be the restrictions of real polynomials to $S^2$; constants, sums, and products show that it is a unital real algebra, and coordinate functions separate distinct points. Fix $f\in C(S^2;\mathbb{R})$ and $\epsilon >0$. For each fixed $\eta \in S^2$, form the family of open sets $U_p:=\{\omega :p(\omega )>f(\omega )-\epsilon \mathrm{/}2\}$ for all $p\in A$ with $p(\eta )=f(\eta )$. This family covers $S^2$: for each $\omega$, [F8] supplies an interpolant $p$ with $p(\eta )=f(\eta )$ and $p(\omega )=f(\omega )$, using the constant polynomial when $\omega =\eta$. By [F9], compactness gives a finite subcover corresponding to $p_1,\dots ,p_s$. Their maximum $g_{\eta }\in \bar{A}$ by [F3] satisfies $g_{\eta }(\eta )=f(\eta )$ and $g_{\eta }>f-\epsilon \mathrm{/}2$ everywhere. This proves the existence of such a function for every $\eta$ without choosing a family indexed by $S^2$. Define the set $\mathcal{G}:=\{g\in \bar{A}:g>f-\epsilon \mathrm{/}2\text{ on }{S}^2\}$ and the open sets $V_g:=\{\omega :g(\omega )<f(\omega )+\epsilon \mathrm{/}2\}$ for $g\in \mathcal{G}$. The functions whose existence was just proved show that these sets cover $S^2$, so [F9] gives a finite subcover $V_{g_1},\dots ,V_{g_t}$. The minimum $h:={\min }_i{g}_i$ lies in $\bar{A}$ by [F3] and satisfies $f-\epsilon \mathrm{/}2<h<f+\epsilon \mathrm{/}2$ everywhere. Since $\epsilon$ was arbitrary, $f\in \bar{A}$ and the real polynomial restrictions are dense in $C(S^2;\mathbb{R})$. Applying this to real and imaginary parts gives density of complex polynomial restrictions in $C(S^2;\mathbb{C})$. By [F4] and [F9], $d\sigma$ is Radon and $C(S^2;\mathbb{C})=C_c(S^2;\mathbb{C})$ is dense in $L^2(S^2,d\sigma ;\mathbb{C})$. Step 2.2 therefore implies that the span of the mutually orthogonal spaces ${\mathcal{H}}_{\mathrm{\ell }}$ is dense in $L^2$.

For fixed $\eta$, rotations fixing $\eta$ leave $Z_{\mathrm{\ell }}(\omega ,\eta )$ invariant. Writing an invariant polynomial in coordinates parallel and perpendicular to $\eta$ shows it is a polynomial in $t=\omega \cdot \eta$ on $S^2$. Since it belongs to ${\mathcal{H}}_{\mathrm{\ell }}$, the spherical-Laplacian formula in the Definition gives $(1-t^2)p^{\mathrm{\prime }\mathrm{\prime }}-2t{p}^{\mathrm{\prime }}+\mathrm{\ell }(\mathrm{\ell }+1)p=0$ for this polynomial $p$. If $p(t)={\sum }_{j=0}^{\mathrm{\ell }}{c}_j{t}^j$, coefficient comparison gives $(j+2)(j+1)c_{j+2}+[\mathrm{\ell }(\mathrm{\ell }+1)-j(j+1)]c_j=0$ for $0\le j<\mathrm{\ell }$; downward recursion determines every coefficient from $c_{\mathrm{\ell }}$, and a nonzero solution cannot have degree $d<\mathrm{\ell }$ because its leading coefficient would be multiplied by $\mathrm{\ell }(\mathrm{\ell }+1)-d(d+1)\mathrm{\ne }0$. Thus polynomial solutions of degree at most $\mathrm{\ell }$ form a one-dimensional space of degree exactly $\mathrm{\ell }$. By step 2.3, $p(1)=Z_{\mathrm{\ell }}(\eta ,\eta )=(2\mathrm{\ell }+1)\mathrm{/}(4\pi )>0$. Consequently $P_{\mathrm{\ell }}(t):=4\pi p(t)\mathrm{/}(2\mathrm{\ell }+1)$ is the unique degree-$\mathrm{\ell }$ solution with $P_{\mathrm{\ell }}(1)=1$ from the Definition, and $Z_{\mathrm{\ell }}(\omega ,\eta )=\frac{2\mathrm{\ell }+1}{4\pi }{P}_{\mathrm{\ell }}(\omega \cdot \eta )$.