Spectrum of the Laplacian on a sphere
= Spectrum of the Laplacian on a sphere
{title2=$\lambda_\ell=\ell(\ell+d-1)$}
For the unit $S^d$ and the nonnegative <Laplace-Beltrami operator>, the <eigenvalues> are $\ell(\ell+d-1)$, with multiplicities $\binom{\ell+d}{d}-\binom{\ell+d-2}{d}$ for $\ell\geq0$. The <eigenfunctions> are restrictions of degree-$\ell$ <harmonic polynomials>. The <harmonic decomposition of homogeneous polynomials> and the <Stone-Weierstrass theorem> prove completeness.