Spherical harmonics: orthogonal decomposition, complete basis and addition theorem

For a homogeneous $Q$ of degree $d$, differentiating $t\mapsto Q(tx)$ at $t=1$ gives Euler's identity $x\cdot \mathrm{\nabla }Q=dQ$ by [F2]; consequently, for homogeneous $Q$ of degree $l-2j$, the product and chain rules and $\mathrm{\Delta }(r^{2j})=2j(2j+1)r^{2j-2}$, $\mathrm{\nabla }(r^{2j})\cdot \mathrm{\nabla }Q=2j(l-2j)r^{2j-2}Q$ give $\mathrm{\Delta }(r^{2j}Q)=2j(2l-2j+1)r^{2j-2}Q+r^{2j}\mathrm{\Delta }Q$.

Continuous functions are dense in $L^2(S^2,d\mathrm{\Omega })$: cover $S^2$ by its six coordinate graph charts, take a subordinate finite ambient partition and smooth cutoffs by [F4], pull a localised $L^2$ function back to the compact planar chart support (the surface density is bounded above and below there, so the pullback is $L^2$ for Lebesgue measure), extend by zero, mollify in the plane by [F5], multiply by the cutoff, pull back and sum the finitely many parts, obtaining continuous functions converging in $L^2(S^2,d\mathrm{\Omega })$.

Fix $x\mathrm{\ne }0$ and rotate coordinates so that $x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }=e_3$; write $s=y_3$ and ${\varrho }^2=y_1^2+y_2^2$. Put $A_j={\pi }^{-1}{\int }_0^{\pi }{\cos }^{2j}\theta d\theta$. Finite expansion of [F7]'s integral and cancellation of the odd cosine powers under $\theta \mapsto \pi -\theta$ give $\mathrm{\mid }x{\mathrm{\mid }}^{l+1}{c}_l(x,y)={\sum }_{j=0}^{\lfloor l\mathrm{/}2\rfloor }{b}_j{s}^{l-2j}{\varrho }^{2j}$ with $b_j=(-1{)}^j\left(\genfrac{}{}{0px}{}{l}{2j}\right)A_j$. This polynomial agrees with the supplied coefficient for $y\mathrm{\ne }0$ and defines its value at zero. Here $A_0=1$; integrating the derivative of $\sin \theta {\cos }^{2j+1}\theta$ by [F8], whose endpoint values vanish, gives $(2j+2)A_{j+1}=(2j+1)A_j$. Thus for $0\le j<\lfloor l\mathrm{/}2\rfloor$, $4(j+1{)}^2{b}_{j+1}=-(l-2j)(l-2j-1)b_j$. Direct coordinate differentiation gives $\mathrm{\Delta }(s^{l-2j}{\varrho }^{2j})=(l-2j)(l-2j-1)s^{l-2j-2}{\varrho }^{2j}+4j^2{s}^{l-2j}{\varrho }^{2j-2}$, with any term having zero coefficient omitted; the displayed recursion cancels every Laplacian coefficient, including the last term because its $s$ exponent is zero or one. Therefore the solid coefficient is homogeneous of degree $l$ and harmonic in $y$. It is nonzero and invariant under rotations fixing $x$, since $b_0=1$ and its value at $y=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ after multiplication by $\mathrm{\mid }x{\mathrm{\mid }}^{l+1}$ is one. Rotation covariance transfers the conclusion to every fixed $x\mathrm{\ne }0$.

For $P\in {\mathcal{P}}_l$ take $H={\sum }_{j=0}^{\lfloor l\mathrm{/}2\rfloor }{a}_j{r}^{2j}{\mathrm{\Delta }}^{j}P$ with $a_0=1$ and $a_j=-a_{j-1}\mathrm{/}[2j(2l-2j+1)]$; applying step 1.1 to each term, the coefficient of $r^{2j}{\mathrm{\Delta }}^{j+1}P$ in $\mathrm{\Delta }H$ is $a_j+2(j+1)(2l-2j-1)a_{j+1}=0$ for $0\le j<\lfloor l\mathrm{/}2\rfloor$, and the remaining top term ${\mathrm{\Delta }}^{\lfloor l\mathrm{/}2\rfloor +1}P$ vanishes because $\mathrm{\Delta }$ lowers degree by two and $2(\lfloor l\mathrm{/}2\rfloor +1)>l$; hence $H\in H_l$ and $P-H\in r^2{\mathcal{P}}_{l-2}$, so ${\mathcal{P}}_l=H_l+r^2{\mathcal{P}}_{l-2}$.

For homogeneous harmonic $H_l,H_m$ with $l\mathrm{\ne }m$, the second Green identity [F3] on the ball $B_R$ with both Laplacians zero gives $0={\int }_{S_R}(H_l{\mathrm{\partial }}_{\nu }{H}_m-H_m{\mathrm{\partial }}_{\nu }{H}_l)=\frac{m-l}{R}{\int }_{S_R}{H}_l{H}_{m}dS$, and ${\int }_{S_R}{H}_l{H}_{m}dS=R^{l+m+2}{\int }_{S^2}{H}_l{H}_{m}d\mathrm{\Omega }$ by homogeneity and [F4]; since $m\mathrm{\ne }l$, the spherical functions $H_l{\mathrm{\mid }}_{S^2}$ and $H_m{\mathrm{\mid }}_{S^2}$ are orthogonal in $L^2(S^2,d\mathrm{\Omega })$, and by Euler the same computation shows the radial derivative of any homogeneous harmonic of degree $k$ on $S_R$ is $k{H}_k\mathrm{/}R$.

Directness is purely algebraic. On real polynomials define the positive inner product $⟨{\sum }_{\alpha }{p}_{\alpha }{x}^{\alpha },{\sum }_{\alpha }{q}_{\alpha }{x}^{\alpha }{⟩}_F={\sum }_{\alpha }\alpha !p_{\alpha }{q}_{\alpha }$, with finite sums and $\alpha !={\alpha }_1!{\alpha }_2!{\alpha }_3!$. Comparing monomial coefficients gives $⟨x_{i}P,Q{⟩}_F=⟨P,{\mathrm{\partial }}_{i}Q{⟩}_F$, hence $⟨r^{2}P,Q{⟩}_F=⟨P,\mathrm{\Delta }Q{⟩}_F$. If a harmonic homogeneous polynomial $H$ equals $r^{2}Q$, then $⟨H,H{⟩}_F=⟨Q,\mathrm{\Delta }H{⟩}_F=0$, so $H=0$. The decomposition of step 2.1 is therefore direct in ZF, and multiplication by $r^2$ is injective (a product with $r^2$ vanishing as a polynomial vanishes at every nonzero point, and hence everywhere). Counting the degree-$l$ monomials gives $\dim H_l=\dim {\mathcal{P}}_l-\dim {\mathcal{P}}_{l-2}=2l+1$, including $l=0,1$ when ${\mathcal{P}}_{l-2}=\{0\}$.

The restricted real polynomial algebra on $S^2$ is unital and separates points (the three coordinate functions separate distinct points of the sphere), so it is uniformly dense in $C(S^2)$ by Stone-Weierstrass [F5]; decomposing each polynomial into homogeneous parts and each homogeneous part by step 2.1 into harmonic pieces of degrees $\le$ its degree, every polynomial restriction lies in the finite linear span of the spaces of spherical harmonics, so that span is dense in $C(S^2)$ in the uniform norm and hence in $L^2(S^2,d\mathrm{\Omega })$ together with step 1.2.

For each $l$ the restrictions of $H_l$ form a finite-dimensional inner-product space of dimension $2l+1$ by step 3.2, and finite Gram-Schmidt [F5] applied to a fixed ordered monomial-elimination basis of $H_l$ produces an orthonormal basis $Y_{l1},\dots ,Y_{l,2l+1}$; by step 3.1 the spaces of different degrees are mutually orthogonal, so the union is an orthonormal family whose closed span contains all spherical harmonics and is dense by step 4.1, that is, it is a complete orthonormal system of $L^2(S^2,d\mathrm{\Omega })$ in the sense of [F5]'s definition.

Fix $l$ and set $K_l(n,n^{\mathrm{\prime }})={\sum }_{m=1}^{2l+1}{Y}_{lm}(n)Y_{lm}(n^{\mathrm{\prime }})$; if ${\tilde{Y}}_{lm}={\sum }_k{O}_{mk}{Y}_{lk}$ is another real orthonormal basis, then ${\sum }_m{\tilde{Y}}_{lm}(n){\tilde{Y}}_{lm}(n^{\mathrm{\prime }})={\sum }_{k,k^{\mathrm{\prime }}}\left({\sum }_m{O}_{mk}{O}_{m{k}^{\mathrm{\prime }}}\right)Y_{lk}(n)Y_{l{k}^{\mathrm{\prime }}}(n^{\mathrm{\prime }})=K_l(n,n^{\mathrm{\prime }})$ because $O$ is orthogonal, so $K_l$ is basis independent.

For a proper rotation $R$ of ${\mathbb{R}}^3$, the map $H\mapsto H\circ R$ permutes the harmonic polynomials of each degree and preserves the sphere measure [F4]; hence $K_l(Rn,R{n}^{\mathrm{\prime }})=K_l(n,n^{\mathrm{\prime }})$. For fixed $n^{\mathrm{\prime }}$, the function $n\mapsto K_l(n,n^{\mathrm{\prime }})$ is the restriction of a homogeneous harmonic polynomial of degree $l$ (a linear combination of the chosen representatives of the $Y_{lm}$) that is invariant under all rotations fixing $n^{\mathrm{\prime }}$; after rotating $n^{\mathrm{\prime }}$ to $e_3$, such an invariant homogeneous polynomial has the form ${\sum }_{j=0}^{\lfloor l\mathrm{/}2\rfloor }{b}_j{x}_3^{l-2j}(x_1^2+x_2^2{)}^j$, because in the complex coordinates $u=x_1+i{x}_2$, $v=x_1-i{x}_2$ the monomial $u^p{v}^q{z}^c$ acquires the factor $e^{i(p-q)\theta }$ under the rotation by $\theta$ and the vanishing of all nonzero Fourier coefficients [F6] forces $p=q$; harmonicity then gives the recursion $4(j+1{)}^2{b}_{j+1}=-(l-2j)(l-2j-1)b_j$, so all coefficients are determined by $b_0$ and the space of invariant harmonics of degree $l$ is at most one dimensional.

By step 1.3 the solid zonal coefficient $c_l(x,y)$ of the convergent kernel lemma, read for fixed $x\mathrm{\ne }0$ as a function of $y$, is a homogeneous harmonic polynomial of degree $l$, equal to $\mathrm{\mid }x{\mathrm{\mid }}^{-l-1}\mathrm{\mid }y{\mathrm{\mid }}^l{P}_l(x\cdot y\mathrm{/}(\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }))$; for $n^{\mathrm{\prime }}=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ and $y=rn$ it restricts to the nonzero invariant harmonic $r^l{P}_l(n\cdot n^{\mathrm{\prime }})$, so by the one-dimensionality of step 7.1 there is a scalar $\lambda (n^{\mathrm{\prime }})$ with $K_l(n,n^{\mathrm{\prime }})=\lambda (n^{\mathrm{\prime }})P_l(n\cdot n^{\mathrm{\prime }})$; since $K_l$ is rotation invariant by step 7.1 and proper rotations act transitively on $S^2$, $\lambda$ is independent of $n^{\mathrm{\prime }}$, and $K_l(n,n)=\lambda P_l(1)=\lambda$ by $P_l(1)=1$.

Integrate the diagonal identity $K_l(n,n)={\sum }_m{Y}_{lm}(n{)}^2=\lambda$ over $S^2$: orthonormality gives ${\sum }_m\mathrm{\parallel }{Y}_{lm}{\mathrm{\parallel }}_{L^2(S^2)}^2=2l+1$ while the constant $\lambda$ integrates to $4\pi \lambda$ by the total mass [F4], so $\lambda =(2l+1)\mathrm{/}(4\pi )$ and ${\sum }_m{Y}_{lm}(n)Y_{lm}(n^{\mathrm{\prime }})=\frac{2l+1}{4\pi }{P}_l(n\cdot n^{\mathrm{\prime }})$; this proves the addition theorem and completes the decomposition, orthogonality, completeness and exact kernel assertions of the Statement, with Countable Choice used only through the cited surface, $L^2$ and mollification interfaces.