Spherical harmonic modes are orthogonal, complete for continuous data, and have polynomial evaluation bounds

For a homogeneous polynomial $Q_n$ of degree $n$, direct differentiation and Euler's identity give $\mathrm{\Delta }(\mathrm{\mid }x{\mathrm{\mid }}^{2j}{Q}_n)=2j(2n+2j+1)\mathrm{\mid }x{\mathrm{\mid }}^{2j-2}{Q}_n+\mathrm{\mid }x{\mathrm{\mid }}^{2j}\mathrm{\Delta }{Q}_n$. For homogeneous $P$ of degree $n$, set $H=P+{\sum }_{j=1}^{\lfloor n\mathrm{/}2\rfloor }{a}_j\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j}P$, where $a_0=1$ and $a_{j+1}=-a_j\mathrm{/}[2(j+1)(2n-2j-1)]$. Substitution into the displayed identity makes the coefficient of each $\mathrm{\mid }x{\mathrm{\mid }}^{2j}{\mathrm{\Delta }}^{j+1}P$ equal to $a_j+2(j+1)(2n-2j-1)a_{j+1}=0$; the last Laplacian power vanishes by degree. Thus $H$ is homogeneous harmonic and $P-H$ is divisible by $\mathrm{\mid }x{\mathrm{\mid }}^2$. Induction on degree shows that restrictions of polynomials are finite sums of elements of the spaces ${\mathcal{H}}_j$.

The polar-coordinate identity $\mathrm{\Delta }={\mathrm{\partial }}_{rr}+2r^{-1}{\mathrm{\partial }}_r+r^{-2}{\mathrm{\Delta }}_{{\mathbb{S}}^2}$ applied to $r^{\mathrm{\ell }}Y(\omega )$, whose extension is harmonic by [F1], gives ${\mathrm{\Delta }}_{{\mathbb{S}}^2}Y=-\mathrm{\ell }(\mathrm{\ell }+1)Y$. On the sphere, integration by parts gives $⟨v,{\mathrm{\Delta }}_{{\mathbb{S}}^2}w⟩=⟨{\mathrm{\Delta }}_{{\mathbb{S}}^2}v,w⟩$: in $(\theta ,\varphi )$ coordinates integrate the $\varphi$ derivative over its period and the $\theta$ term with weight $\sin \theta$; the pole boundary terms vanish because smooth functions and their spherical gradients are bounded while $\sin \theta \to 0$. Distinct degrees have distinct eigenvalues, so their spaces are orthogonal.

Choose a finite orthonormal basis from [F2] and write $g={\sum }_m{c}_m{Y}_{\mathrm{\ell }m}$. Cauchy–Schwarz gives $\mathrm{\mid }g(\omega ){\mathrm{\mid }}^2\le ({\sum }_m\mathrm{\mid }{c}_m{\mathrm{\mid }}^2)({\sum }_m\mathrm{\mid }{Y}_{\mathrm{\ell }m}(\omega ){\mathrm{\mid }}^2)$. Orthonormality identifies the first sum with $\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$, and [F2] gives the second as $(2\mathrm{\ell }+1)\mathrm{/}(4\pi )$. Taking the supremum proves the evaluation bound.

Real polynomial restrictions contain constants and separate distinct points of ${\mathbb{S}}^2$ by a coordinate function. Compactness and [F3] make them dense in real $C({\mathbb{S}}^2)$; applying this separately to real and imaginary parts gives density in complex $C({\mathbb{S}}^2)$. By step 1.1 this is density of the algebraic sum of the ${\mathcal{H}}_j$. For continuous $f$ and $\epsilon >0$, choose a finite sum $q$ of harmonic modes with $\mathrm{\parallel }f-q{\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$. Put $P_{\le N}f={\sum }_{\mathrm{\ell }=0}^N{P}_{\mathrm{\ell }}f$ and $V_N={⨁}_{\mathrm{\ell }=0}^N{\mathcal{H}}_{\mathrm{\ell }}$. Orthogonality from step 1.2 makes $f-P_{\le N}f$ orthogonal to $V_N$, so Pythagoras shows $P_{\le N}f$ is a best $L^2$ approximation in $V_N$. For $N$ at least the largest degree in $q$, $\mathrm{\parallel }f-P_{\le N}f{\mathrm{\parallel }}_2\le \mathrm{\parallel }f-q{\mathrm{\parallel }}_2\le \sqrt{4\pi }\epsilon$. Since $\epsilon$ is arbitrary, the ordered partial sums converge in $L^2$.

For smooth $f$ and $\mathrm{\ell }\ge 1$, self-adjointness just proved gives $P_{\mathrm{\ell }}({\mathrm{\Delta }}_{{\mathbb{S}}^2}^{m}f)=(-\mathrm{\ell }(\mathrm{\ell }+1){)}^m{P}_{\mathrm{\ell }}f$. The finite-dimensional projection is a contraction in $L^2$, hence $\mathrm{\parallel }{P}_{\mathrm{\ell }}f{\mathrm{\parallel }}_2\le [\mathrm{\ell }(\mathrm{\ell }+1){]}^{-m}\mathrm{\parallel }{\mathrm{\Delta }}_{{\mathbb{S}}^2}^{m}f{\mathrm{\parallel }}_2$. Taking $m$ arbitrarily large proves the asserted decay (and degree zero is absorbed into $C_m(f)$).

Let $X_1,X_2,X_3$ be the infinitesimal rotation fields on the sphere about the coordinate axes. Rotations preserve ${\mathcal{H}}_{\mathrm{\ell }}$ and surface measure, so $X_j$ preserves ${\mathcal{H}}_{\mathrm{\ell }}$ and is skew-adjoint. Expanding these three fields in spherical coordinates gives $X_1^2+X_2^2+X_3^2={\mathrm{\Delta }}_{{\mathbb{S}}^2}$. Thus ${\sum }_j\mathrm{\parallel }{X}_{j}g{\mathrm{\parallel }}_2^2=-⟨{\mathrm{\Delta }}_{{\mathbb{S}}^2}g,g⟩=\mathrm{\ell }(\mathrm{\ell }+1)\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$, so each $X_j$ has operator norm at most $\sqrt{\mathrm{\ell }(\mathrm{\ell }+1)}$ on ${\mathcal{H}}_{\mathrm{\ell }}$. Iteration and step 1.3 bound every word of $q$ rotation derivatives pointwise by $\sqrt{(2\mathrm{\ell }+1)\mathrm{/}(4\pi )}[\mathrm{\ell }(\mathrm{\ell }+1){]}^{q\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_2$. A finite smooth coordinate cover of the compact sphere expresses each coordinate derivative through order $p$ as bounded smooth combinations of words in the $X_j$ of order at most $p$; compactness supplies a finite cover by [F3], and absorbing its chart bounds gives the stated $C_p(1+\mathrm{\ell }{)}^{p+1\mathrm{/}2}$ estimate.